Mat116 Appendices Student Solutions

A. FACTORING AND LCMS

B. FRACTION NOTATION

C. EXPONENTIAL NOTATION AND ORDER OF OPERATIONS

D. REVIEW OF FACTORING POLYNOMIALS

E. INTRODUCTORY ALGEBRA REVIEW

F. HANDLING DIMENSION SYMBOLS

G. MEAN, MEDIAN, AND MODE

H. SYNTHETIC DIVISION

I. DETERMINANTS AND CRAMER’S RULE

J. ELIMINATION USING MATRICES

K. THE ALGEBRA OF FUNCTIONS

L. DISTANCE, MIDPOINTS, AND CIRCLES

Appendixes

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Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley.Copyright ©2007 by Pearson Education, Inc.

940

APPENDIX A: Factoring and LCMs

Factors and Prime Factorizations

We begin our review with factoring, which is a necessary skill for addition andsubtraction with fraction notation. Factoring is also an important skill in algebra. You will eventually learn to factor algebraic expressions.

The numbers we will be factoring are natural numbers:

1, 2, 3, 4, 5, and so on.

To factor a number means to express the number as a product. Considerthe product We say that 3 and 4 are factors of 12 and that is afactorization of 12. Since we also know that 12 and 1 are factorsof 12 and that is a factorization of 12.

EXAMPLE 1 Find all the factors of 12.

We first find some factorizations:

The factors of 12 are 1, 2, 3, 4, 6, and 12.

EXAMPLE 2 Find all the factors of 150.

We first find some factorizations:

The factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, and 150.

Note that the word “factor” is used both as a noun and as a verb. You factor when you express a number as a product. The numbers you multiplytogether to get the product are factors.

Do Exercises 1–4 (in the margin at left).

PRIME NUMBER

A natural number that has exactly two different factors, itself and 1, iscalled a prime number.

EXAMPLE 3 Which of these numbers are prime? 7, 4, 11, 18, 1

7 is prime. It has exactly two different factors, 1 and 7.

4 is not prime. It has three different factors, 1, 2, and 4.

11 is prime. It has exactly two different factors, 1 and 11.

18 is not prime. It has factors 1, 2, 3, 6, 9, and 18.

1 is not prime. It does not have two different factors.

150 � 2 � 5 � 3 � 5.150 � 10 � 15,150 � 6 � 25,

150 � 5 � 30,150 � 3 � 50,150 � 2 � 75,150 � 1 � 150,

12 � 2 � 2 � 3.12 � 3 � 4,12 � 2 � 6,12 � 1 � 12,

12 � 112 � 12 � 1,

3 � 412 � 3 � 4.

AA FACTORING AND LCMS

Find all the factors of the number.

ObjectivesFind all the factors ofnumbers and find primefactorizations of numbers.

Find the LCM of two or morenumbers using primefactorizations.

1. 9 2. 16

3. 24 4. 180

Answers on page A-62

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In the margin at right is a table of the prime numbers from 2 to 157. Thereare more extensive tables, but these prime numbers will be the most helpfulto you in this text.

Do Exercise 5.

If a natural number, other than 1, is not prime, we call it composite. Everycomposite number can be factored into a product of prime numbers. Such afactorization is called a prime factorization.

EXAMPLE 4 Find the prime factorization of 36.

We begin by factoring 36 any way we can. One way is like this:

The factors 4 and 9 are not prime, so we factor them:

The factors in the last factorization are all prime, so we now have the primefactorization of 36. Note that 1 is not part of this factorization because it is not prime.

Another way to find the prime factorization of 36 is like this:

In effect, we begin factoring any way we can think of and keep factoring untilall factors are prime. Using a factor tree might also be helpful.

36

2 18

3 6

2 3

36 � 2 � 3 � 2 � 3

36 � 2 � 18 � 2 � 3 � 6 � 2 � 3 � 2 � 3.

� 2 � 2 � 3 � 3

36 � 4 � 9

36 � 4 � 9.

5. Which of these numbers are prime?

8, 6, 13, 14, 1

Find the prime factorization.

6. 48

7. 50

8. 770


941


or

36

4 9

2 2 3 3

36 � 2 � 2 � 3 � 3

or

36

3 12

2 6

2 3

36 � 3 � 2 � 2 � 3

No matter which way we begin, the result is the same: The prime factori-zation of 36 contains two factors of 2 and two factors of 3. Every compositenumber has a unique prime factorization.

EXAMPLE 5 Find the prime factorization of 60.

This time, we use the list of primes from the table. We go through the tableuntil we find a prime that is a factor of 60. The first such prime is 2.

We keep dividing by 2 until it is not possible to do so.

Now we go to the next prime in the table that is a factor of 60. It is 3.

Each factor in is a prime. Thus this is the prime factorization.

Do Exercises 6–8.

2 � 2 � 3 � 5

60 � 2 � 2 � 3 � 5

60 � 2 � 2 � 15

60 � 2 � 30

2, 3, 5, 7, 11, 13, 17,19, 23, 29, 31, 37, 41,43, 47, 53, 59, 61, 67,71, 73, 79, 83, 89, 97,101, 103, 107, 109, 113, 127, 131, 137,139, 149, 151, 157

A TABLE OF PRIMES

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Least Common Multiples

Least common multiples are used to add and subtract with fraction notation.The multiples of a number all have that number as a factor. For example,

the multiples of 2 are

2, 4, 6, 8, 10, 12, 14,

We could name each of them in such a way as to show 2 as a factor. For example,

The multiples of 3 all have 3 as a factor:

3, 6, 9, 12, 15,

Two or more numbers always have many multiples in common. Fromlists of multiples, we can find common multiples.

EXAMPLE 6 Find the common multiples of 2 and 3.

We make lists of their multiples and circle the multiples that appear inboth lists.

2, 4, 6 , 8, 10, 12 , 14, 16, 18 , 20, 22, 24 , 26, 28, 30 , 32, 34,

3, 6 , 9, 12 , 15, 18 , 21, 24 , 27, 30 , 33,

The common multiples of 2 and 3 are

6, 12, 18, 24, 30,

Do Exercises 9 and 10.

In Example 6, we found common multiples of 2 and 3. The least, or small-est, of those common multiples is 6. We abbreviate least common multipleas LCM.

There are several methods that work well for finding the LCM of severalnumbers. Some of these do not work well in algebra, especially when we consider expressions with variables such as 4ab and 12abc. We now review a method that will work in arithmetic and in algebra as well. To see how it works, let’s look at the prime factorizations of 9 and 15 in order to find the LCM:

Any multiple of 9 must have two 3’s as factors. Any multiple of 15 must haveone 3 and one 5 as factors. The smallest multiple of 9 and 15 is

Two 3’s; 9 is a factor

One 3, one 5; 15 is a factor

The LCM must have all the factors of 9 and all the factors of 15, but the factorsare not repeated when they are common to both numbers.

To find the LCM of several numbers using prime factorizations:

a) Write the prime factorization of each number.b) Form the LCM by writing the product of the different factors from

step (a), using each factor the greatest number of times that itoccurs in any one of the factorizations.

3 � 3 � 5 � 45.

15 � 3 � 5.9 � 3 � 3,

36, . . . .

36 , . . . .

36 , . . . ;

18, . . . .

14 � 2 � 7.

16, . . . .

9. Find the common multiples of 3 and 5 by making lists ofmultiples.

10. Find the common multiples of 9 and 15 by making lists ofmultiples.


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EXAMPLE 7 Find the LCM of 40 and 100.

a) We find the prime factorizations:

b) The different prime factors are 2 and 5. We write 2 as a factor three times(the greatest number of times that it occurs in any one factorization). Wewrite 5 as a factor two times (the greatest number of times that it occurs inany one factorization).

The LCM is or 200.


EXAMPLE 8 Find the LCM of 27, 90, and 84.

a) We factor:

b) We write 2 as a factor two times, 3 three times, 5 one time, and 7 one time.

The LCM is or 3780.

Do Exercise 13.


Since 7 is prime, it has no prime factorization. It still, however, must be afactor of the LCM:

The LCM is or 21.

If one number is a factor of another, then the LCM is the larger of thetwo numbers.



We have

The LCM is or 72.

If two or more numbers have no common prime factor, then the LCMis the product of the numbers.


2 � 2 � 2 � 3 � 3,

9 � 3 � 3.

8 � 2 � 2 � 2,

7 � 3,

21 � 3 � 7.

7 � 7,

2 � 2 � 3 � 3 � 3 � 5 � 7,

84 � 2 � 2 � 3 � 7.

90 � 2 � 3 � 3 � 5,

27 � 3 � 3 � 3,

2 � 2 � 2 � 5 � 5,

100 � 2 � 2 � 5 � 5.

40 � 2 � 2 � 2 � 5,

Find the LCM by factoring.

11. 8 and 10

12. 18 and 27

13. Find the LCM of 18, 24, and 30.

Find the LCM.

14. 3, 18

15. 12, 24

Find the LCM.

16. 4, 9

17. 5, 6, 7


943


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Co

pyr

igh

t ©

200

7 P

ears

on

Ed

uca

tio

n, I

nc.

AA EXERCISE SET

Find all the factors of the number.

1. 20 2. 36 3. 72 4. 81

Find the prime factorization of the number.

5. 15 6. 14 7. 22 8. 33 9. 9

10. 25 11. 49 12. 121 13. 18 14. 24

15. 40 16. 56 17. 90 18. 120 19. 210

20. 330 21. 91 22. 143 23. 119 24. 221

Find the prime factorization of the numbers. Then find the LCM.

25. 4, 5 26. 18, 40 27. 24, 36 28. 24, 27 29. 3, 15

30. 20, 40 31. 30, 40 32. 50, 60 33. 13, 23 34. 12, 18

35. 18, 30 36. 45, 72 37. 30, 36 38. 30, 50 39. 24, 30

40. 60, 70 41. 17, 29 42. 18, 24 43. 12, 28 44. 35, 45

45. 2, 3, 5 46. 3, 5, 7 47. 24, 36, 12 48. 8, 16, 22

49. 5, 12, 15 50. 12, 18, 40 51. 6, 12, 18 52. 24, 35, 45

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APPENDIX B: Fraction Notation

We now review fraction notation and its use with addition, subtraction,multiplication, and division of arithmetic numbers.

Equivalent Expressions and Fraction Notation

An example of fraction notation for a number is

The top number is called the numerator, and the bottom number is called thedenominator.

The whole numbers consist of the natural numbers and 0:

The arithmetic numbers, also called the nonnegative rational numbers,consist of the whole numbers and the fractions, such as and The arith-metic numbers can also be described as follows.

ARITHMETIC NUMBERS

The arithmetic numbers are the whole numbers and the fractions,such as 8, and All these numbers can be named with fractionnotation where a and b are whole numbers and

Note that all whole numbers can be named with fraction notation. For ex-ample, we can name the whole number 8 as We call 8 and equivalentexpressions.

Being able to find an equivalent expression is critical to a study of algebra.Some simple but powerful properties of numbers that allow us to find equiva-lent expressions are the identity properties of 0 and 1.

THE IDENTITY PROPERTY OF 0(ADDITIVE IDENTITY)

For any number a,

(Adding 0 to any number gives that same number—for example,)

THE IDENTITY PROPERTY OF 1(MULTIPLICATIVE IDENTITY)

For any number a,

Multiplying any number by 1 gives that same number—for example, 35 � 1 � 3

5 .��

a � 1 � a.

12 � 0 � 12.

a � 0 � a.

81

81 .

b � 0.ab ,

65 .3

4 ,

95 .2

3

0, 1, 2, 3, 4, 5, . . . .

23

BB FRACTION NOTATION ObjectivesFind equivalent fractionexpressions by multiplying by 1.

Simplify fraction notation.

Add, subtract, multiply, anddivide using fractionnotation.

Numerator

Denominator

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Here are some ways to name the number 1:

The following property allows us to find equivalent fraction expressions, thatis, find other names for arithmetic numbers.

EQUIVALENT EXPRESSIONS FOR 1

For any number a,

We can use the identity property of 1 and the preceding result to findequivalent fraction expressions.

EXAMPLE 1 Write a fraction expression equivalent to with a denomina-tor of 15.

Note that We want fraction notation for that has a denomi-nator of 15, but the denominator 3 is missing a factor of 5. We multiply by 1,using as an equivalent expression for 1. Recall from arithmetic that to mul-tiply with fraction notation, we multiply numerators and denominators:

Using the identity property of 1

Using for 1

Multiplying numerators and denominators

Do Exercises 1–3.

Simplifying Expressions

We know that and so on, all name the same number. Any arithmeticnumber can be named in many ways. The simplest fraction notation is thenotation that has the smallest numerator and denominator. We call theprocess of finding the simplest fraction notation simplifying. We reversethe process of Example 1 by first factoring the numerator and the denomina-tor. Then we factor the fraction expression and remove a factor of 1 using theidentity property of 1.

EXAMPLE 2 Simplify:

Factoring the fraction expression

Using the identity property of 1 (removing a factor of 1) �23

�23

� 1

�23

�55

1015

�2 � 53 � 5

1015

.

48 ,2

4 ,12 ,

�1015

.

55

�23

� 55

23

�23

� 1

55

2315 � 3 � 5.

23

aa

� 1.

a � 0,

55

,33

, and2626

.

Factoring the numerator and the denominator. In thiscase, each is the prime factorization.

1. Write a fraction expressionequivalent to with adenominator of 12.

2. Write a fraction expressionequivalent to with adenominator of 28.

3. Multiply by 1 to find threedifferent fraction expressionsfor


78 .

34

23

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EXAMPLE 3 Simplify:

Factoring the numerator and the denominator

Factoring the fraction expression

Removing a factor of 1

It is always a good idea to check at the end to see if you have indeed fac-tored out all the common factors of the numerator and the denominator.

CANCELINGCanceling is a shortcut that you may have used to remove a factor of 1 whenworking with fraction notation. With great concern, we mention it as a pos-sible way to speed up your work. You should use canceling only when remov-ing common factors in numerators and denominators. Each common factorallows us to remove a factor of 1 in a product. Canceling cannot be donewhen adding. Our concern is that “canceling” be performed with care andunderstanding. Example 3 might have been done faster as follows:

Caution!

The difficulty with canceling is that it is often applied incorrectly insituations like the following:

Wrong! Wrong! Wrong!

The correct answers are

In each situation, the number canceled was not a factor of 1. Factors areparts of products. For example, in 2 and 3 are factors, but in 2and 3 are not factors. Canceling may not be done when sums ordifferences are in numerators or denominators, as shown here.

Do Exercises 4–6.

2 � 3,2 � 3,

2 � 32

�52

;4 � 14 � 2

�56

;1554

�5

18.

2 � 32

� 3;4 � 14 � 2

�12

;1554

�14

.

3624

�2 � 3 � 2 � 32 � 2 � 3 � 2

�32

, or3624

�3 � 122 � 12

�32

, or3618

2412

�

3

2

32

.

�32

� 1 �32

�2 � 3 � 22 � 3 � 2

�32

3624

�2 � 3 � 2 � 32 � 2 � 3 � 2

3624

.Simplify.

4.

5.

6.


7227

3818

1845

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We can always insert the number 1 as a factor. The identity property of 1allows us to do that.

EXAMPLE 4 Simplify:

or

EXAMPLE 5 Simplify:

Removing a factor of 1:

Simplifying


Multiplication, Addition, Subtraction, and Division

After we have performed an operation of multiplication, addition, subtrac-tion, or division, the answer may or may not be in simplified form. We sim-plify, if at all possible.

MULTIPLICATIONTo multiply using fraction notation, we multiply the numerators to get thenew numerator, and we multiply the denominators to get the newdenominator.

MULTIPLYING FRACTIONS

To multiply fractions, multiply the numerators and multiply thedenominators:

EXAMPLE 6 Multiply and simplify:


Factoring the numerator and the denominator


Simplifying


�3

10

3 � 53 � 5

� 1 �3 � 5 � 3

3 � 5 � 2 � 5

�5 � 3 � 3

2 � 3 � 5 � 5

56

�9

25�

5 � 96 � 25

56

�9

25.

ab

�cd

�a � cb � d

.

�81

� 8

99

� 1 �8 � 91 � 9

Factoring and inserting a factorof 1 in the denominator

729

�8 � 91 � 9

729

.

1872

�1 � 184 � 18

�14

1872

�2 � 98 � 9

�28

�2 � 12 � 4

�14

,

1872

.

Simplify.

7.

8.

Multiply and simplify.

9.

10.


38

�53

�72

65

�2512

4812

2754

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ADDITIONWhen denominators are the same, we can add by adding the numerators andkeeping the same denominator.

ADDING FRACTIONS WITH LIKE DENOMINATORS

To add fractions when denominators are the same, add thenumerators and keep the same denominator:

EXAMPLE 7 Add:

The common denominator is 8. We add the numerators and keep thecommon denominator:

In arithmetic, we generally write as (See a review of converting froma mixed numeral to fraction notation at left.) In algebra, you will find that im-proper fraction symbols such as are more useful and are quite proper for ourpurposes.

What do we do when denominators are different? We find a common de-nominator. We can do this by multiplying by 1. Consider adding and Thereare several common denominators that can be obtained. Let’s look at twopossibilities.

A.

Simplifying �1112

�2224

�4

24�

1824

�16

�44

�34

�66

16

�34

�16

� 1 �34

� 1

34 .1

6

98

1 18 .9

8

48

�58

�4 � 5

8�

98

.

48

�58

.

ac

�bc

�a � b

c.

B.

�1112

�2

12�

912

�16

�22

�34

�33

16

�34

�16

� 1 �34

� 1

We had to simplify in (A). We didn’t have to simplify in (B). In (B), we usedthe least common multiple of the denominators, 12. That number is calledthe least common denominator, or LCD.

ADDING FRACTIONS WITH DIFFERENT DENOMINATORS

To add fractions when denominators are different:

a) Find the least common multiple of the denominators. Thatnumber is the least common denominator, LCD.

b) Multiply by 1, using an appropriate notation, to express eachfraction in terms of the LCD.

c) Add the numerators, keeping the same denominator.d) Simplify, if possible.

n�n,

To convert from a mixednumeral to fraction notation:

�a Multiply the wholenumber by thedenominator:

�b Add the result to thenumerator:

�c Keep the denominator.

24 � 5 � 29.

3 � 8 � 24.

3

58

�298�a

�b�c

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950


EXAMPLE 8 Add and simplify:

The LCM of the denominators, 8 and 12, is 24. Thus the LCD is 24. Wemultiply each fraction by 1 to obtain the LCD:

EXAMPLE 9 Add and simplify:

We first look for the LCM of 30 and 18. That number is then the LCD. Wefind the prime factorization of each denominator:

The LCD is or 90. To get the LCD in the first denominator, we needa factor of 3. To get the LCD in the second denominator, we need a factor of 5.We get these numbers by multiplying by 1:

Multiplying by 1

Simplifying

Do Exercises 11–14.

SUBTRACTIONWhen subtracting, we also multiply by 1 to obtain the LCD. After we havemade the denominators the same, we can subtract by subtracting the numer-ators and keeping the same denominator.

EXAMPLE 10 Subtract and simplify:

The LCD is 40.

�45 � 32

40�

1340

�4540

�3240

98

�45

�98

�55

�45

�88

98

�45

.

�2945

.

�2 � 29

5 � 2 � 3 � 3

�58

5 � 2 � 3 � 3

�33

5 � 2 � 3 � 3�

252 � 3 � 3 � 5

1130

�5

18�

115 � 2 � 3

�33

�5

2 � 3 � 3�

55

5 � 2 � 3 � 3,

1130

�5

18�

115 � 2 � 3

�5

2 � 3 � 3.

1130

�5

18.

�1924

.

�9 � 10

24

�9

24�

1024

38

�5

12�

38

�33

�5

12�

22

38

�5

12.

Multiplying by 1. Since wemultiply the first number by Since

we multiply the secondnumber by 22 .2 � 12 � 24,

33 .

3 � 8 � 24,

Adding the numerators and keeping thesame denominator

The denominators are nowthe LCD.

Adding the numerators andkeeping the LCD

Factoring the numerator andremoving a factor of 1

Subtracting the numerators andkeeping the same denominator

Add and simplify.

11.

12.

13.

14.


1324

�7

40

56

�7

10

56

�76

45

�35

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951


EXAMPLE 11 Subtract and simplify:

The LCD is 10.



RECIPROCALSTwo numbers whose product is 1 are called reciprocals, or multiplicative inverses, of each other. All the arithmetic numbers, except zero, havereciprocals.

EXAMPLES

12. The reciprocal of is because

13. The reciprocal of 9 is because

14. The reciprocal of is 4 because


RECIPROCALS AND DIVISIONReciprocals and the number 1 can be used to justify a fast way to divide arith-metic numbers. We multiply by 1, carefully choosing the expression for 1.

EXAMPLE 15 Divide by

This is a symbol for 1.


Simplifying

After multiplying, we had a denominator of or 1. That was because weused the reciprocal of the divisor, for both the numerator and the denomi-nator of the symbol for 1.

Do Exercise 21.

57 ,

3535 ,

�1021

3535 � 1 �

10213535

�1021

1

�23 �

57

75 �

57

23

�75

�2375

�2375

�

5757

75

.23

14 � 4 � 4

4 � 1.14

9 �19 � 9

9 � 1.19

23 �

32 � 6

6 � 1.32

23

55

� 1 �1 � 52 � 5

�12

�5

10

�7

10�

210

�7 � 2

10

7

10�

15

�7

10�

15

�22

710

�15

.

Multiplying by We use because it is the

reciprocal of 75 .

57

5757

.

Subtract and simplify.

15.

16.

Find the reciprocal.

17.

18.

19. 5

20.

21. Divide by multiplying by 1:


3547

.

13

157

411

512

�29

78

�25

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When multiplying by 1 to divide, we get a denominator of 1. What do weget in the numerator? In Example 15, we got This is the product of thedividend, and the reciprocal of the divisor.

DIVIDING FRACTIONS

To divide fractions, multiply by the reciprocal of the divisor:

EXAMPLE 16 Divide by multiplying by the reciprocal of the divisor:

is the reciprocal of

Multiplying

After dividing, always simplify if possible.

EXAMPLE 17 Divide and simplify:

is the reciprocal of









33

� 1

24 �38

�241

�38

�241

�83

�24 � 81 � 3

�3 � 8 � 8

1 � 3�

8 � 81

� 64

24 �38

.

55

� 1

56

� 30 �56

�301

�56

�1

30�

5 � 16 � 30

�5 � 1

6 � 5 � 6�

16 � 6

�1

36

56

� 30.

�32

2 � 32 � 3

� 1 �2 � 3 � 33 � 2 � 2

�2 � 93 � 4

49

94

23

�49

�23

�94

23

�49

.

�56

35

53

12

�35

�12

�53

12 �

35 .

ab

�cd

�ab

�dc

.

57 ,

23 ,2

3 �57 .

⎫⎪⎪⎬⎪⎪⎭

⎫⎪⎪⎬⎪⎪⎭

Divide by multiplying by thereciprocal of the divisor. Thensimplify.

22.

23.

24.

Divide and simplify.

25.

26.


36 �49

78

� 56

295

12

54

�32

43

�72

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Write an equivalent expression for each of the following. Use the indicated name for 1.

953

Exercise Set B

EXERCISE SETBB

1. 2. 3. 4.89

�Use 44

for 1.�35

�Use 2020

for 1.�56

�Use 1010

for 1.��Use 33

for 1.�34

Write an equivalent expression with the given denominator.

5. (Denominator: 24) 6. (Denominator: 54)29

78

Simplify.

7. 8. 9. 10. 11. 12.13

1046

424827

5614

4956

1827

13. 14. 15. 16. 17. 18.15025

10020

1751

1976

13211

567

19. 20. 21. 22. 23. 24.13 � v39 � v

8 � x6 � x

48001600

26001400

625325

425525

Compute and simplify.

25. 26. 27. 28.1011

�1110

154

�34

1516

�85

13

�14

29. 30. 31. 32.98

�7

123

10�

815

14

�13

49

�1318

33. 34. 35. 36.1516

�5

121112

�38

125

�25

54

�34

41. 42. 43. 44.56

� 15100 �15

17638

89

�4

15

37. 38. 39. 40.1

20�

15

76

�35

1516

�23

1112

�25

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954

APPENDIX C: Exponential Notation and Order of Operations

CC EXPONENTIAL NOTATION AND ORDER OF OPERATIONSObjectives

Write exponential notationfor a product.

Evaluate exponentialexpressions.

Simplify expressions usingthe rules for order ofoperations.

Exponential Notation

Exponents provide a shorter way of writing products. An abbreviation for aproduct in which the factors are the same is called a power. For

we write

3 factors

This is read “ten to the third power.” We call the number 3 an exponent andwe say that 10 is the base. An exponent of 2 or greater tells how many timesthe base is used as a factor. For example,

In this case, the exponent is 4 and the base is a. An expression for a power iscalled exponential notation.

This is the exponent.

This is the base.

EXAMPLE 1 Write exponential notation for .

Do Exercises 1–3.

Evaluating Exponential Expressions

EXAMPLE 2 Evaluate:

EXAMPLE 3 Evaluate:

We have

We could also carry out the calculation as follows:

EXPONENTIAL NOTATION

For any natural number n greater than or equal to 2,

n factors

Do Exercises 4–6.

bn � b � b � b � b � � � b.

34 � 3 � 3 � 3 � 3 � 9 � 3 � 3 � 27 � 3 � 81.

34 � 3 � 3 � 3 � 3 � 9 � 9 � 81.

34.

52 � 5 � 5 � 25

52.

10 � 10 � 10 � 10 � 10 � 105

10 � 10 � 10 � 10 � 10

an

a � a � a � a � a 4.

103.10 � 10 � 10,⎫⎪⎬⎪⎭

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

Write exponential notation.

1.

2.

3.

Evaluate.

4.

5.

6.


�1.1�3

83

104

1.08 � 1.08

6 � 6 � 6 � 6 � 6

4 � 4 � 4

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Order of Operations

What does mean? If we add 4 and 5 and multiply the result by 2, weget 18. If we multiply 5 and 2 and add 4 to the result, we get 14. Since the re-sults are different, we see that the order in which we carry out operations isimportant. To indicate which operation is to be done first, we use groupingsymbols such as parentheses or brackets or braces For example,

but Grouping symbols tell us what to do first. If there are no grouping

symbols, we have agreements about the order in which operations should be done.

RULES FOR ORDER OF OPERATIONS

1. Do all calculations within grouping symbols before operationsoutside.

2. Evaluate all exponential expressions.3. Do all multiplications and divisions in order from left to right.4. Do all additions and subtractions in order from left to right.

EXAMPLE 4 Calculate:

Multiplying

Subtracting

Adding


Always calculate within parentheses first. When there are exponents andno parentheses, simplify powers before multiplying or dividing.


Working within parentheses first

Evaluating the exponential expression


Evaluating the exponential expression

Multiplying

Note that


There are no parentheses,so we find 42 first.

Multiplying

Adding

Subtracting


� 78

� 94 � 16

� 7 � 87 � 16

7 � 3 � 29 � 42 � 7 � 3 � 29 � 16

7 � 3 � 29 � 42.

�3 � 4�2 � 3 � 42.

� 48

3 � 42 � 3 � 16

3 � 42.

� 144

�3 � 4�2 � �12�2

�3 � 4�2.

� 8

� 5 � 3

15 � 2 � 5 � 3 � 15 � 10 � 3

15 � 2 � 5 � 3.

3 � �5 � 6� � 3 � 11 � 33.�3 � 5� � 6 � 15 � 6 � 21,� �.� �,� �,

4 � 5 � 2

Calculate.

7.

8.

Calculate.

9.

10.

11.

12.


8 � 2 � 53 � 4 � 20

2 � 53

�2 � 5�3

18 � 4 � 3 � 7

4 � 5 � 2

16 � 3 � 5 � 4

955


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Doing the divisions in order fromleft to right

Doing the second division


Doing the division first

Multiplying

Multiplying


Sometimes combinations of grouping symbols are used, as in

The rules still apply. We begin with the innermost grouping symbols—in thiscase, the parentheses—and work to the outside.


Subtracting within the parentheses first

Adding inside the brackets

Multiplying

A fraction bar can play the role of a grouping symbol.


An equivalent expression with brackets as grouping symbols is

What this shows, in effect, is that we do the calculations first in the numera-tor and then in the denominator, and then divide the results:


12�9 � 7� � 4 � 533 � 24 �

12�2� � 4 � 527 � 16

�24 � 20

11�

4411

� 4.

�12�9 � 7� � 4 � 5� � �33 � 24�.

12�9 � 7� � 4 � 533 � 24 .

� 50

� 5�10� 5�4 � �8 � 2�� 5�4 � 6�

5�4 � �8 � 2��.

5�4 � �8 � 2��.

� 8000

� 10,000 �45

1000 �1

10�

45

� �1000 � 10� �45

1000 �1

10 �45 .

� 4

2.56 � 1.6 � 0.4 � 1.6 � 0.4

2.56 � 1.6 � 0.4.Calculate.

13.

14.

Calculate.

15.

16.


13�10 � 6� � 4 � 9

52 � 32

4��8 � 3� � 7�

1000 �1

10�

45

51.2 � 0.64 � 40

956


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957

Exercise Set C

EXERCISE SETCCWrite exponential notation.

1. 2. 3. 10 � 10 � 103 � 3 � 3 � 3 � 35 � 5 � 5 � 5

4. 5. 6. 18 � 1810 � 10 � 10 � 10 � 10 � 101 � 1 � 1

Evaluate.

7. 8. 9. 10. 11. 102124954372

12. 13. 14. 15. 16. �0.1�3�2.3�2�1.8�21415

22. 23. 24. 25. 26. 2000 � �1.06�21000 � �1.02�3�1.4�35324

17. 18. 19. 20. 21. � 38�2� 4

5�2

�20.4�2�14.8�2�0.2�3

Calculate.

27. 28. 29. 30. 30�5� � 2�2�9�8� � 7�6�14 � 6 � 69 � 2 � 8

31. 32. 33. 34. 32 � 8 � 4 � 29 � 3 � 16 � 814 � 2 � 6 � 739 � 4 � 2 � 2

35. 36. 37. 38. 3 � 23�6 � 3�2�5 � 4�27 � 10 � 10 � 2

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958


Co

pyr

igh

t ©

200

7 P

ears

on

Ed

uca

tio

n, I

nc.

39. 40. 41. 42. 7 � 22�8 � 2�3�7 � 3�24 � 52

43. 44. 45. 46. 10 � 32�3 � 2�2�5 � 2�26 � 42

47. 48. 49. 50. 7 � 34 � 18120 � 33 � 4 � 620 � 43 � 8 � 443 � 8 � 4

51. 52. 53. 54. �8 � 7� � 98 � �7 � 9�8��13 � 6� � 11�6�9 � �3 � 4��

55. 56. 57. 58. �12 � 8� � 412 � �8 � 4�15 � 4 � 15 � 215�4 � 2�

59. 60. 61. 62. 400 � 0.64 � 3.22000 �3

50�

32

256 � 32 � 41000 � 100 � 10

67. 68. 69.3�6 � 7� � 5 � 46 � 7 � 8�4 � 1�

52 � 43 � 3

92 � 22 � 1580 � 62

92 � 32

70. 71. 72. 95 � 23 � 5 � �24 � 4�8 � 2 � �12 � 0� � 3 � �5 � 2�20�8 � 3� � 4�10 � 3�10�6 � 2� � 2�5 � 2�

63. 64. 84 � 12 � 10 � 35 � 8 � 2 � 1675 � 15 � 4 � 8 � 32

65. 66. 20 � 45 � 15 � 15 � 60 � 1216 � 5 � 80 � 12 � 36 � 9

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Factor. Check by multiplying.

1.

2.

Factor.

3.

4.


24 � 11t � t2

m2 � 8m � 12

y2 � 7y � 10

x2 � 5x � 6

959

APPENDIX D: Review of Factoring Polynomials

Factoring Trinomials:

CONSTANT TERM POSITIVERecall the FOIL method* of multiplying two binomials:

OFF O I L

I

L

The product is a trinomial. In this example, the leading term has a coefficientof 1. The constant term is positive. To factor we think of FOIL inreverse.

EXAMPLE 1 Factor:

Think of FOIL in reverse. The first term of each factor is x. We are lookingfor numbers p and q such that

We look for two numbers p and q whose product is 8 and whose sum is 9.Since both 8 and 9 are positive, we need consider only positive factors.

The factorization is We can check by multiplying:


EXAMPLE 2 Factor:

Since the constant term, 20, is positive and the coefficient of the middleterm, �9, is negative, we look for a factorization of 20 in which both factorsare negative. Their sum must be �9.

The factorization is � y � 4� � y � 5�.

y2 � 9y � 20.

�x � 1� �x � 8� � x2 � 9x � 8.

�x � 1� �x � 8�.

x2 � 9x � 8 � �x � p� �x � q� � x2 � � p � q�x � pq.

x2 � 9x � 8.

x2 � 8x � 15,

� x2 � 8x � 15.

�x � 3� �x � 5� � x2 � 5x � 3x � 15

x 2 � bx � c

DD REVIEW OF FACTORINGPOLYNOMIALS

ObjectivesFactor trinomials of the type

Factor trinomials of the type, , by the

FOIL method.

Factor trinomial squares.

Factor differences of squares.

Factor sums and differencesof cubes.

a � 1ax

2 � bx � c

x

2 � bx � c.

*For a review of the ac-method, see Section 5.4.

⎫⎪⎬⎪⎭

2, 4 6

1, 8 9

PAIRS OF FACTORS SUMS OF FACTORS

The numbers we need are1 and 8.

�1, �20 �21

�2, �10 �12

�4, �5 �9


The numbers we need are�4 and �5.

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Do Exercises 3 and 4 on the preceding page.

CONSTANT TERM NEGATIVE

EXAMPLE 3 Factor:

Always look first for the largest common factor. This time x is the com-mon factor. We first factor it out:

Now consider Since the constant term, is negative, we lookfor a factorization of in which one factor is positive and one factor is nega-tive. The sum of the factors must be the coefficient of the middle term, sothe negative factor must have the larger absolute value. Thus we consider onlypairs of factors in which the negative factor has the larger absolute value.

The factorization of is But do not forget the com-mon factor! The factorization of the original trinomial is

Do Exercises 5–7.

EXAMPLE 4 Factor:

Since the constant term, �110, is negative, we look for a factorization of�110 in which one factor is positive and one factor is negative. Their summust be 17, so the positive factor must have the larger absolute value.

The factorization is


�x � 5� �x � 22�.

x2 � 17x � 110.

x�x � 5� �x � 6�.

�x � 5� �x � 6�.x2 � x � 30

�1,�30

�30,x2 � x � 30.

x3 � x2 � 30x � x�x2 � x � 30�.

x3 � x2 � 30x.

5. a) Factor

b) Explain why you would notconsider these pairs of factorsin factoring

Factor.

6.

7.

Factor.

8.

9.

10.


x2 � 110 � x

y2 � 4y � 12

x3 � 4x2 � 12x

2x3 � 2x2 � 84x

x3 � 3x2 � 54x

x2 � x � 20.

x2 � x � 20.

960


1, �30 �29

2, �15 �13

3, �10 �7

5, �6 �1


The numbers we want are5 and �6.

�1, 110 109

�2, 55 53

�5, 22 17

�10, 11 1


The numbers we need are�5 and 22.

We consider only pairs of factorsin which the positive term hasthe larger absolute value.

1, 20

2, 10

4, 5

�1, �20

�2, �10

�4, �5

PAIRS OF FACTORSPRODUCTS

OF FACTORS

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Factoring Trinomials:

We consider the FOIL method for factoring trinomials of the type

, .

Consider the following multiplication.

F O I L

To factor , we must reverse what we just did. We look for twobinomials whose product is this trinomial. The product of the First termsmust be . The product of the Outside terms plus the product of the Insideterms must be 23x. The product of the Last terms must be 10. We know fromthe preceding discussion that the answer is . In general,however, finding such an answer involves trial and error. We use the follow-ing method.

THE FOIL METHOD

To factor trinomials of the type using the FOILmethod:

1. Factor out the largest common factor.2. Find two First terms whose product is :

FOIL

3. Find two Last terms whose product is c:

FOIL

4. Repeat steps (2) and (3) until a combination is found for which thesum of the Outside and Inside products is bx :

I FOILO

5. Always check by multiplying.

EXAMPLE 5 Factor: .

1. First, we factor out the largest common factor, if any. There is none (otherthan 1 or �1).

2. Next, we factor the first term, . The only possibility is . The desired factorization is then of the form .

3. We then factor the last term, �8, which is negative. The possibilities are, , , and . They can be written in either order.��2� �4�2��4�8��1��8� �1�

�� x ��3x �

3x � x3x2

3x2 � 10x � 8

� � ax2 � bx � c.x �� x ��

� � ax2 � bx � c.� � x �� x �

x � � � ax2 � bx � c.x � � ��

ax2

ax2 � bx � c, a � 1,

�3x � 2� �4x � 5�

12x2

12x2 � 23x � 10

� 12x2 � 23x � 10

�3x � 2� �4x � 5� � 12x2 � 15x � 8x � 10

a � 1ax2 � bx � c

a � 1ax 2 � bx � c,

961


⎫⎪⎬⎪⎭

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4. We look for combinations of factors from steps (2) and (3) such that thesum of the outside and the inside products is the middle term, 10x :

3x

;

�8x Wrong middle term

�12x

;

2x Wrong middle term

�3x � 2� �x � 4� � 3x2 � 10x � 8

�3x � 8� �x � 1� � 3x2 � 5x � 8

Factor by the FOIL method.

11.

12.


3x2 � 5x � 2

3x2 � 13x � 56

962


�3x

;

8x Wrong middle term

12x

�2x Correct middle term!

�3x � 2� �x � 4� � 3x2 � 10x � 8

�3x � 8� �x � 1� � 3x2 � 5x � 8

There are four other possibilities that we could try, but we have a fac-torization: .

5. Check : .


EXAMPLE 6 Factor: .

1. First, we factor out the largest common factor, if any. The expression is common to all terms, so we factor it out: .

2. Next, we factor the trinomial . We factor the first term, , and get or We then have these as possibilities for fac-

torizations: or .

3. We then factor the last term, 10, which is positive. The possibilities areand They can be written in either

order.

4. We look for combinations of factors from steps (2) and (3) such that thesum of the outside and the inside products is the middle term, �19x. Thesign of the middle term is negative, but the sign of the last term, 10, ispositive. Thus the signs of both factors of the last term, 10, must be nega-tive. From our list of factors in step (3), we can use only �10, �1 and �5,�2 as possibilities. This reduces the possibilities for factorizations byhalf. We begin by using these factors with . Should wenot find the correct factorization, we will consider .

�3x

;


�6x

;


�3x � 5� �2x � 2� � 6x2 � 16x � 10

�3x � 10� �2x � 1� � 6x2 � 23x � 10

�� x ��6x �

�� 2x ��3x �

��5� ��2�.�5� �2�,��10� ��1�,�10� �1�,

�� x ��6x �� 2x ��3x �

3x � 2x.6x � x,6x26x2 � 19x � 10

3x4�6x2 � 19x � 10�3x4

18x6 � 57x5 � 30x4

�3x � 2� �x � 4� � 3x2 � 10x � 8

�3x � 2� �x � 4�

�30x

;


�15x

�4x Correct middle term!

�3x � 2� �2x � 5� � 6x2 � 19x � 10

�3x � 1� �2x � 10� � 6x2 � 32x � 10

We have a correct answer. We need not consider .The factorization of is . But do not

forget the common factor! We must include it in order to get a completefactorization of the original trinomial:

.18x6 � 57x5 � 30x4 � 3x4�3x � 2� �2x � 5�

�3x � 2� �2x � 5�6x2 � 19x � 10�� x ��6x �

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5. Check :


Next, we consider some special factoring methods. When we recognizecertain types of polynomials, we can factor more quickly using these spe-cial methods. Most of them are the reverse of the methods of special multiplication.

Trinomial Squares

Consider the trinomial . We look for factors of 9 whose sum is 6.We see that these factors are 3 and 3 and the factorization is

.

Note that the result is the square of a binomial. We also call a trinomial square, or perfect-square trinomial.

The factors of a trinomial square are two identical binomials. We use thefollowing equations.

TRINOMIAL SQUARES

;

EXAMPLE 7 Factor: .

We find the square terms and write their square roots with a minus signbetween them.Note the sign!

EXAMPLE 8 Factor: .

Rewriting in descending order

We find the square terms andwrite their square roots with aplus sign between them.


Differences of Squares

The following are differences of squares:

, , .

To factor a difference of two expressions that are squares, we can use a patternfor multiplying a sum and a difference that we used earlier.

a2 � 49b249 � 4y2x2 � 9

� �4y � 7�2

16y2 � 49 � 56y � 16y2 � 56y � 49

16y2 � 49 � 56y

x2 � 10x � 25 � �x � 5�2

x2 � 10x � 25

A2 � 2AB � B2 � �A � B�2

A2 � 2AB � B2 � �A � B�2

x2 � 6x � 9

x2 � 6x � 9 � �x � 3� �x � 3� � �x � 3�2

x2 � 6x � 9

� 18x6 � 57x5 � 30x4.

3x4�3x � 2� �2x � 5� � 3x4�6x2 � 19x � 10� Factor.

13.

14.

Factor.

15.

16.


9y2 � 30y � 25

x2 � 14x � 49

20x5 � 46x4 � 24x3

24y2 � 46y � 10

963


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DIFFERENCE OF TWO SQUARES

To factor a difference of two squares, write the square root of the firstexpression plus the square root of the second times the square root ofthe first minus the square root of the second.

EXAMPLE 9 Factor: .

EXAMPLE 10 Factor: .


Sums or Differences of Cubes

We can factor the sum or the difference of two expressions that are cubes.Consider the following products:

and

.

The above equations (reversed) show how we can factor a sum or a differenceof two cubes.

SUM OR DIFFERENCE OF CUBES

;

Note that what we are considering here is a sum or a difference of cubes. We are not cubing a binomial. For example, is not the same as . The table of cubes in the margin is helpful.


We have

.

In one set of parentheses, we write the cube root of the first term, x. Then we write the cube root of the second term, �10. This gives us the expres-sion

.�x � 10� � �

x � 10:

x3 � 1000 � x3 � 103

A3 � B3

x3 � 1000

A3 � B3�A � B�3

A3 � B3 � �A � B� �A2 � AB � B2�A3 � B3 � �A � B� �A2 � AB � B2�

� A3 � B3

� A3 � A2B � AB2 � A2B � AB2 � B3

�A � B� �A2 � AB � B2� � A�A2 � AB � B2� � B�A2 � AB � B2�

� A3 � B3

� A3 � A2B � AB2 � A2B � AB2 � B3

�A � B� �A2 � AB � B2� � A�A2 � AB � B2� � B�A2 � AB � B2�

x2 �1

16 � x2 � �14�2 � �x �

14� �x �

14�

x2 �1

16

x2 � 9 � x2 � 32 � �x � 3� �x � 3�

A2 � B2� �A � B� �A � B�

x2 � 9

A2 � B2 � �A � B� �A � B�

Factor.

17.

18.


m2 �19

y2 � 4

964


0.2 0.008

0.1 0.001

0 0

1 1

2 8

3 27

4 64

5 125

6 216

7 343

8 512

9 729

10 1000

N N

3

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To get the next factor, we think of and do the following:

Square the first term: .

Multiply the terms, , and thenchange the sign: 10x.

Square the second term: .

.

Note that we cannot factor . It is not a trinomial square norcan it be factored by trial and error. Check this on your own.



We have

.

In one set of parentheses, we write the cube root of the first term, 5. Then wewrite a plus sign, and then the cube root of the second term, 2y :

.

To get the next factor, we think of and do the following:

Square the first term: .

Multiply the terms, , and thenchange the sign: �10y.

Square the second term: .

.


FACTORING SUMMARY

Trinomial squares: ;

Sum of cubes: ;

Difference of cubes: ;

Difference of squares: ;

Sum of squares: cannot be factored using realnumbers if the largest common factor has been removed.

A2 � B2

A2 � B2 � �A � B� �A � B�A3 � B3 � �A � B� �A2 � AB � B2�A3 � B3 � �A � B� �A2 � AB � B2�A2 � 2AB � B2 � �A � B�2A2 � 2AB � B2 � �A � B�2

�A � B� � A2 � AB � B2�

�5 � 2y� �25 � 10y � 4y2�

2y � 2y � 4y2

5 � 2y � 10y

5 � 5 � 25

5 � 2y

�5 � 2y� � �

125 � 8y3 � 53 � �2y�3

125 � 8y3

x2 � 10x � 100

�A � B� �A2 � AB � B2�

�x � 10��x2 � 10x � 100�

��10�2 � 100

x��10� � �10x

x � x � x2

x � 10 Factor.

19.

20.

Factor.

21.

22.


8 � 64y3

x3 � 1000

1 � x3

w 3 � 27

965


⎫ ⎬ ⎭

⎫ ⎬ ⎭

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Factor.

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DD EXERCISE SET

1. 2. 3. t 2 � 15 � 2tx2 � 8x � 33y2 � 10y � 21

4. 5. 6. p2 � 3p � 542a3 � 20a2 � 50a2y3 � 16y2 � 32y

7. 8. 9. 10y � y2 � 2412x � x2 � 27m2 � m � 72

10. 11. 12. t 2 � 4t � 3p2 �25

p �1

25y2 �

23

y �19

Factor.

13. 14. 15. 16. 6x3 � x2 � 12x10y3 � y2 � 21y8x2 � 6x � 93x2 � 14x � 5

17. 18. 19. 20. 9a2 � 18a � 835y2 � 34y � 812b2 � 8b � 13c2 � 20c � 32

21. 22. 23. 24. 18x2 � 24 � 6x8x2 � 16 � 28x8x � 30x2 � 64t � 10t 2 � 6

25. 26. 27. 28. 70x4 � 68x3 � 16x214x4 � 19x3 � 3x215x3 � 19x2 � 10x12x3 � 31x2 � 20x

29. 30. 31. 32. 6y2 � y � 29x2 � 15x � 46a2 � 7a � 103a2 � a � 4

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967

Exercise Set D

Factor.

33. 34. 35. y2 � 18y � 81y2 � 16y � 64x2 � 4x � 4

36. 37. 38. x2 � 1 � 2xx2 � 1 � 2xx2 � 8x � 16

39. 40. 41. �18y2 � y3 � 81y25x2 � 60x � 369y2 � 12y � 4

42. 43. 44. 20y2 � 100y � 12512a2 � 36a � 2724a2 � a3 � 144a

Factor.

45. 46. 47. p2 � 49y2 � 9x2 � 16

48. 49. 50. 9x3 � 25x4a3 � 49am2 � 64

51. 52. 1100 � y21

36 � z2

Factor.

53. 54. 55. 56. c3 � 64x3 � 1a3 � 8z3 � 27

61. 62. 63. 64. 54x3 � 224a3 � 33z3 � 32y3 � 128

57. 58. 59. 60. 64 � 125x38 � 27b327x3 � 18a3 � 1

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968

APPENDIX E: Introductory Algebra Review

This text is appropriate for a two-semester course that combines thestudy of introductory and intermediate algebra. Students who take only thesecond-semester course (which generally begins with Chapter 7) often need areview of the topics covered in the first-semester course. This appendix is aguide for a review of the first six chapters of this text. Below is a syllabus ofselected exercises that can be used as a condensed review of the main objec-tives in the first half of the text. For extra help, consult the Student’s SolutionsManual, which contains fully worked-out solutions with step-by-step anno-tations for all the odd-numbered exercises in the exercise sets.

EE INTRODUCTORY ALGEBRA REVIEW

1.3a 5–13 3, 11, 19, 25, 33, 39, 43

1.4a 6–12 5, 9, 15, 19, 21, 51, 55, 69, 81

1.5a 4–12, 17 17, 29, 33, 39, 53, 69

1.6c 19–23 49, 53, 55, 57

1.7c 14–17 47, 49, 59

1.7d 28– 31 73, 81, 83

1.7e 32–38 89, 97, 101, 107

1.8b 7, 11, 12 15, 21

1.8c 16 29, 35

1.8d 20–23 41, 51, 61, 81

2.1b 6, 7 19, 39, 47

2.2a 2, 3, 6 3, 7, 27, 33

2.3a 2 5, 15

2.3b 7, 9 23, 43, 51

2.3c 11, 12, 13 71, 73, 83, 87

2.5a 4, 5, 6, 8 7, 15, 25, 35

2.6a 1, 2, 5, 8 1, 7, 15, 27

2.7e 13, 16 53, 61, 75

3.2a 2, 3, 5 5, 13, 21

3.3a 1 21, 25

3.3b 3, 4 45, 51

3.4a 1 9, 11

3.4c 5, 9, 10, 11 37, 51, 55

4.1b 2 15, 17

4.1c 5, 6 27, 35

4.1d, e, f 8, 11, 13, 15, 22–27 67, 71, 83, 91, 95, 103

4.2a, b 1–4, 10, 15, 16 3, 35, 41, 45

4.4a 3, 4 7, 17

4.4c 9, 10 33, 41, 49

4.5d 10 55

4.6a 2–6 9, 29

4.6b 11–13 43, 47

4.6c 16–18 63, 67, 71

4.7c 4–6 21, 27

4.7f 11, 12 41, 57

4.8b 7, 8, 10 25, 29, 33, 39

SECTION/OBJECTIVE EXAMPLES EXERCISES IN EXERCISE SET

(continued)

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Two other features of the text that can be used for review of the first sixchapters are as follows.

• At the end of each chapter is a Summary and Review that provides an ex-tensive set of review exercises. Reference codes beside each exercise or di-rection line preceding it allow the student to easily return to the objectivebeing reviewed. Answers to all of these exercises appear at the back ofthe book.

• The Cumulative Review that follows both Chapter 3 and Chapter 6 canalso be used for review. Each reviews material from all preceding chap-ters. At the back of the text are answers to all Cumulative Review exer-cises, together with section and objective references, so that studentsknow exactly what material to study if they miss an exercise.

The extensive supplements package that accompanies this text alsoincludes material appropriate for a structured review of the first six chapters.Consult the preface in the text for detailed descriptions of each of thefollowing.

• Videotapes and Digital Video Tutor

• Work It Out! Chapter Test Video on CD

• MathXL® Tutorials on CD

• MathXL®

• MyMathLab

969

APPENDIX E: Introductory Algebra Review

5.1b 9–12 19, 27

5.1c 13, 15, 18 39, 47

5.2a 1, 2, 3 1, 21, 27

5.3a 1, 2 3, 11, 25, 39, 71

5.4a 1 19, 35

5.5b 4, 6 13, 19, 29

5.5d 13, 17, 19 51, 55, 63, 77

5.6a 1, 2 1, 9, 23, 27

5.8b 4 25, 29, 37

5.9b 6 25

6.1d 11, 12 57, 59

6.2b 6, 8 13, 31

6.3c 4, 6 25, 31

6.4a 5, 6, 8 11, 17, 27, 59

6.5a 3, 4 9, 17, 39

6.7a 1, 3, 5 5, 11, 21, 35

6.8a 2 13

6.8b 3 27

SECTION/OBJECTIVE EXAMPLES EXERCISES IN EXERCISE SET

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970

APPENDIX F: Handling Dimension Symbols

Calculating with Dimension Symbols

In many applications, we add, subtract, multiply, and divide quantitieshaving units, or dimensions, such as ft, km, sec, and hr. For example, to findaverage speed, we divide total distance by total time. What results is nota-tion very much like a rational expression.

EXAMPLE 1 A car travels 150 km in 2 hr. What is its average speed?

, or

(The standard abbreviation for is , but it does not suit our pres-ent discussion well.)

The symbol makes it look as though we are dividing kilometersby hours. It can be argued that we can divide only numbers. Nevertheless, wetreat dimension symbols, such as km, ft, and hr, as if they were numerals orvariables, obtaining correct results mechanically.

Do Exercise 1.

EXAMPLES Compare the following.

2. with

3. with

4. with (square feet)

Do Exercises 2–4.

If 5 men work 8 hours, the total amount of labor is 40 man-hours.

EXAMPLE 5 Compare

with .

Do Exercise 5.

5 men � 8 hours � 40 man-hours5x � 8y � 40xy

5 ft � 3 ft � 15 ft25x � 3x � 15x2

3 ft � 2 ft � �3 � 2� ft � 5 ft3x � 2x � �3 � 2�x � 5x

150 km2 hr

�150

2kmhr

� 75kmhr

150x2y

�150

2�

xy

� 75xy

km�hr

km�hkm�hr

75kmhr

Speed �150 km

2 hr

FF HANDLING DIMENSION SYMBOLS

1. A truck travels 210 mi in 3 hr.What is its average speed?

Perform these calculations.

2.

3.

4.

5. Calculate: .


6 men � 11 hours

24 in. � 3 in.

7 yd � 9 yd

100 m4 sec

ObjectivesPerform calculations withdimension symbols.

Make unit changes.

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EXAMPLE 6 Compare

with .

If an electrical device uses 300 kW (kilowatts) for 240 hr over a period of 15 days, its rate of usage of energy is 4800 kilowatt-hours per day. The standard abbreviation for kilowatt-hours is kWh.

Do Exercise 6.

Making Unit Changes

We can treat dimension symbols much like numerals or variables, becausewe obtain correct results that way. We can change units by substituting or bymultiplying by 1, as shown below.

EXAMPLE 7 Convert 3 ft to inches.

METHOD 1. We have 3 ft. We know that , so we substitute 12 in.for ft:

METHOD 2. We want to convert from “ft” to “in.” We multiply by 1 using asymbol for 1 with “ft” on the bottom since we are converting from “ft,” andwith “in.” on the top since we are converting to “in.”

Do Exercise 7.

We can multiply by 1 several times to make successive conversions. In thefollowing example, we convert to by converting successivelyfrom to to to .

EXAMPLE 8 Convert 60 to .

.

Do Exercise 8.

�60 � 5280

60 � 60�

mimi

�hrhr

�minmin

�ft

sec� 88

ftsec

60mihr

� 60mihr

�5280 ft

1 mi�

1 hr60 min

�1 min60 sec

ft�secmi�hr

ft�secft�minft�hrmi�hrft�secmi�hr

�3 � 12

1�

ftft

� in. � 36 in.

3 ft � 3 ft �12 in.

1 ft

3 ft � 3 � 12 in. � 36 in.

1 ft � 12 in.

300 kW � 240 hr15 da

� 4800kW-hr

da300x � 240y

15t� 4800

xyt

6. Calculate:

.

7. Convert 7 ft to inches.

8. Convert 90 to .


ft�secmi�hr

200 kW � 140 hr35 da

971


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Add the measures.

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FF EXERCISE SET

1. 2. 3. 4. 3.4 lb � 5.2 lb17 g � 28 g55 km�hr � 27 km�hr45 ft � 23 ft

Find the average speeds, given total distance and total time.

5. 90 mi, 6 hr 6. 640 km, 20 hr 7. 9.9 m, 3 sec 8. 76 ft, 4 min

Perform the calculations.

9. 10. 11. 36 ft �1 yd3 ft

60 men � 8 hr20 da

3 in. � 8 lb6 sec

12. 13. 14.3 lb14 ft

�7 lb6 ft

5 ft3 � 11 ft355mihr

� 4 hr

15. Divide $4850 by 5 days. 16. Divide $25.60 by 8 hr.

Make the unit changes.

17. Change 3.2 lb to oz ( ). 18. Change 6.2 km to m.16 oz � 1 lb

19. Change 35 to . 20. Change $375 per day to dollars per minute.ft�minmi�hr

21. Change 8 ft to in. 22. Change 25 yd to ft.

23. How many years ago is 1 million sec ago? Let.

24. How many years ago is 1 billion sec ago?365 days � 1 yr

25. How many years ago is 1 trillion sec ago? 26. Change 20 lb to oz.

27. Change to . 28. Change to .mihr

44ft

secozin.

60lbft

29. Change 2 days to seconds. 30. Change 128 hr to days.

31. Change 216 to . 32. Change 1440 man-hours to man-days.ft2in2

33. Change to . 34. Change the speed of light, 186,000 , to .mi�yrmi�sectonyd380

lbft3

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Find the mean. Round to the nearest tenth.

1. 28, 103, 39

2. 85, 46, 105.7, 22.1

3. A student scored the followingon five tests:

78, 95, 84, 100, 82.

What was the average score?


973

APPENDIX G: Mean, Median, and Mode

Mean, Median, and Mode

One way to analyze data is to look for a single representative number, called acenter point or measure of central tendency. Those most often used are themean (or average), the median, and the mode.

MEANLet’s first consider the mean, or average.

MEAN, OR AVERAGE

The mean, or average, of a set of numbers is the sum of the numbersdivided by the number of addends.

EXAMPLE 1 Consider the following data on total net revenue, inbillions of dollars, for Starbucks Corporation for the years 2000–2004:

$2.2, $2.6, $3.3, $4.1, $5.3

What is the mean of the numbers?Source: Starbucks Corporation

First we add the numbers:

Then we divide by the number of addends, 5:

The mean, or average, revenue of Starbucks for those five years is $3.5 billion.

Note that If we use this cen-ter point, 3.5, repeatedly as the addend, we get the same sum that wedo when adding the individual data numbers.

Do Exercises 1–3.

MEDIANThe median is useful when we wish to de-emphasize extreme values. For example, suppose five workers in a technology company manufactured thefollowing number of computers during one day’s work:

Sarah: 88 Jen: 94

Matt: 92 Mark: 91

Pat: 66

Let’s first list the values in order from smallest to largest:

66 88 91 92 94.

Middle number

The middle number—in this case, 91—is the median.

3.5 � 3.5 � 3.5 � 3.5 � 3.5 � 17.5.

�2.2 � 2.6 � 3.3 � 4.1 � 5.3�5

�17.5

5� 3.5.

2.2 � 2.6 � 3.3 � 4.1 � 5.3 � 17.5.

GG MEAN, MEDIAN, AND MODEObjectiveFind the mean (average), themedian, and the mode of aset of data and solve relatedapplied problems.

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MEDIAN

Once a set of data has been arranged from smallest to largest, themedian of the set of data is the middle number if there is an oddnumber of data numbers. If there is an even number of datanumbers, then there are two middle numbers and the median is the average of the two middle numbers.

EXAMPLE 2 What is the median of the following set of yearly salaries?

$76,000, $58,000, $87,000, $32,500, $64,800, $62,500

We first rearrange the numbers in order from smallest to largest.

$32,500, $58,000, $62,500, $64,800, $76,000, $87,000

Median

There is an even number of numbers. We look for the middle two, which are $62,500 and $64,800. In this case, the median is the average of $62,500 and $64,800:

Do Exercises 4–6.

MODEThe last center point we consider is called the mode. A number that occursmost often in a set of data can be considered a representative number or center point.

MODE

The mode of a set of data is the number or numbers that occur mostoften. If each number occurs the same number of times, there is no mode.

EXAMPLE 3 Find the mode of the following data:

23, 24, 27, 18, 19, 27

The number that occurs most often is 27. Thus the mode is 27.


83, 84, 84, 84, 85, 86, 87, 87, 87, 88, 89, 90.

There are two numbers that occur most often, 84 and 87. Thus the modesare 84 and 87.


115, 117, 211, 213, 219.

Each number occurs the same number of times. The set of data has no mode.


$62,500 � $64,8002

� $63,650.

Find the median.

4. 17, 13, 18, 14, 19

5. 17, 18, 16, 19, 13, 14

6. 122, 102, 103, 91, 83, 81, 78,119, 88

Find any modes that exist.

7. 33, 55, 55, 88, 55

8. 90, 54, 88, 87, 87, 54

9. 23.7, 27.5, 54.9, 17.2, 20.1

10. In conducting laboratory tests,Carole discovers bacteria indifferent lab dishes grew to the following areas, in squaremillimeters:

25, 19, 29, 24, 28.

a) What is the mean?

b) What is the median?

c) What is the mode?


974

APPENDIX G: Mean, Median, and Mode

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For each set of numbers, find the mean (average), the median, and any modes that exist.

975

Exercise Set G

EXERCISE SETGG

1. 17, 19, 29, 18, 14, 29 2. 72, 83, 85, 88, 92 3. 5, 37, 20, 20, 35, 5, 25

4. 13, 32, 25, 27, 13 5. 4.3, 7.4, 1.2, 5.7, 7.4 6. 13.4, 13.4, 12.6, 42.9

7. 234, 228, 234, 229, 234, 278 8. $29.95, $28.79, $30.95, $29.95

9. Atlantic Storms and Hurricanes. The following bargraph shows the number of Atlantic storms orhurricanes that formed in various months from 1980 to 2000. What is the average number for the 9 monthsgiven? the median? the mode?

10. Cheddar Cheese Prices. The following prices per pound of sharp cheddar cheese were found at fivesupermarkets:

$5.99, $6.79, $5.99, $6.99, $6.79.

What was the average price per pound? the medianprice? the mode?

11

25

60

72

29

15

1 1 1

June July Aug. Sept. Oct. Nov.April May Dec.July Aug. Sept. Oct. Nov.April May Dec.

Source: Colorado State University

Atlantic Storms and HurricanesTropical storm and hurricane formation in 1980–2000, by month

June July Aug. Sept. Oct. Nov.April May Dec.

11. Coffee Consumption. The following lists the annualcoffee consumption, in cups per person, for variouscountries. Find the mean, the median, and the mode.

Germany 1113

United States 610

Switzerland 1215

France 798

Italy 750Source: Beverage Marketing Corporation

12. NBA Tall Men. The following is a list of the heights, ininches, of the tallest men in the NBA in a recent year.Find the mean, the median, and the mode.

Shaquille O’Neal 85

Gheorghe Muresan 91

Shawn Bradley 90

Priest Lauderdale 88

Rik Smits 88

David Robinson 85

Arvydas Sabonis 87Source: National Basketball Association

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976

APPENDIX H: Synthetic Division

Synthetic Division

To divide a polynomial by a binomial of the type we can streamline thegeneral procedure by a process called synthetic division.

Compare the following. In A, we perform a division. In B, we also dividebut we do not write the variables.

A.

29

11x � 22 11x � 7

5x2 � 10x 5x2 � x

4x3 � 8x2 x � 2�4x3 � 3x2 � x � 7

4x2 � 5x � 11

x � a,

HH SYNTHETIC DIVISIONObjectiveUse synthetic division todivide a polynomial by abinomial of the type

.x � a

B.

29

11 � 22 11 � 7

5 � 10 5 � 1

4 � 8 1 � 2�4 � 3 � 1 � 7

4 � 5 � 11

In B, there is still some duplication of writing. Also, since we can subtractby adding the opposite, we can use 2 instead of �2 and then add instead ofsubtracting.

C. Synthetic Division

a) Write the 2, the opposite of �2 in the divisor and the coefficients of the dividend.

Bring down the first coefficient.

b)Multiply 4 by 2 to get 8. Add 8 and �3.

c)Multiply 5 by 2 to get 10. Add 10 and 1.

d)Multiply 11 by 2 to get 22. Add 22 and 7.

Quotient Remainder

The last number, 29, is the remainder. The other numbers are the coefficientsof the quotient with that of the term of highest degree first, as follows. Notethat the degree of the term of highest degree is 1 less than the degree of thedividend.

4 5 11 29 Remainder

Zero-degree coefficient

First-degree coefficient

Second-degree coefficient

4 5 11 29

8 10 22 2 �4 �3 1 7

4 5 11

8 10 2 �4 �3 1 7

4 5

8

2 �4 �3 1 7

4

x � 2, 2 �4 �3 1 7

⎫⎪⎬⎪⎭

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The answer is R 29; or

EXAMPLE 1 Use synthetic division to divide:

We have

The answer is R 0; or just

Do Exercise 1.

When there are missing terms, be sure to write 0’s for their coefficients.

EXAMPLES Use synthetic division to divide.

2.

There is no x-term, so we must write a 0 for its coefficient. Note thatso we write �3 at the left.

The answer is R 4; or

3.

Note that so we write �4 at the left.

The answer is

4.

The divisor is so we write 1 at the left.

The answer is

5.

Note that so we write �2 at the left.

The answer is R �218; or


8x4 � 16x3 � 26x2 � 52x � 105 ��218x � 2

.

8x4 � 16x3 � 26x2 � 52x � 105,

8 �16 26 �52 105 �218

�16 32 �52 104 �210 �2 �8 0 �6 0 1 �8

x � 2 � x � ��2�,�8x5 � 6x3 � x � 8� � �x � 2�

x3 � x2 � x � 1.

1 1 1 1 0

1 1 1 1 1 �1 0 0 0 �1

x � 1,

�x4 � 1� � �x � 1�

x2 � 1.

1 0 �1 0

�4 0 4 �4 �1 4 �1 �4

x � 4 � x � ��4�,�x3 � 4x2 � x � 4� � �x � 4�

2x2 � x � 3 �4

x � 3.2x2 � x � 3,

2 1 �3 4

�6 �3 9 �3 �2 7 0 �5

x � 3 � x � ��3�,

�2x3 � 7x2 � 5� � �x � 3�

x2 � 8x � 15.x2 � 8x � 15,

1 8 15 0

2 16 30

2 �1 6 �1 �30

�x3 � 6x2 � x � 30� � �x � 2�.

4x2 � 5x � 11 �29

x � 2.4x2 � 5x � 11,

1. Use synthetic division to divide:

Use synthetic division to divide.

2.

3.


� y3 � 1� � � y � 1�

�x3 � 2x2 � 5x � 4� � �x � 2�

�2x3 � 4x2 � 8x � 8� � �x � 3�.

977


It is important to rememberthat in order for syntheticdivision to work, the divisormust be of the form thatis, a variable minus a constant.The coefficient of the variablemust be 1.

x � a,

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Use synthetic division to divide.

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HH EXERCISE SET

1. 2. 3. �a2 � 11a � 19� � �a � 4��x3 � 2x2 � 2x � 5� � �x � 1��x3 � 2x2 � 2x � 5� � �x � 1�

4. 5. 6. �x3 � 7x2 � 13x � 3� � �x � 2��x3 � 7x2 � 13x � 3� � �x � 2��a2 � 11a � 19� � �a � 4�

7. 8. 9. � y3 � 3y � 10� � � y � 2��3x3 � 7x2 � 4x � 3� � �x � 3��3x3 � 7x2 � 4x � 3� � �x � 3�

10. 11. 12. �6y4 � 15y3 � 28y � 6� � � y � 3��3x4 � 25x2 � 18� � �x � 3��x3 � 2x2 � 8� � �x � 2�

13. 14. 15. � y4 � 16� � � y � 2�� y3 � 125� � � y � 5��x3 � 8� � �x � 2�

16. �x5 � 32� � �x � 2�

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Evaluate.

1.

2.


� 5�1

�2�1�

�34 21�

979

APPENDIX I: Determinants and Cramer’s Rule

In Chapter 8, you probably noticed that the elimination method concernsitself primarily with the coefficients and constants of the equations. Here welearn a method for solving a system of equations using just the coefficientsand constants. This method involves determinants.

Evaluating Determinants

The following symbolism represents a second-order determinant:

.

To evaluate a determinant, we do two multiplications and subtract.

EXAMPLE 1 Evaluate:

.

We multiply and subtract as follows:

.

Determinants are defined according to the pattern shown in Example 1.

SECOND-ORDER DETERMINANT

The determinant is defined to mean .

The value of a determinant is a number. In Example 1, the value is 44.


Third-Order Determinants

A third-order determinant is defined as follows.

Note that the a’s come from the first column.

a1

a2

a3

b1

b2

b3

c1

c2

c3

� a1 �b2

b3

c2

c3� � a2 �b1

b3

c1

c3� � a3 �b1

b2

c1

c2�

a1b2 � a2b1�a1

a2

b1

b2�

�26 �57� � 2 � 7 � 6 � ��5� � 14 � 30 � 44

�26 �57�

�a1

a2

b1

b2�

II DETERMINANTS AND CRAMER’S RULE

ObjectivesEvaluate second-orderdeterminants.

Evaluate third-orderdeterminants.

Solve systems of equationsusing Cramer’s rule.

Note the minus sign here.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

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Note that the second-order determinants on the right can be obtained bycrossing out the row and the column in which each a occurs.

For : For :

For :

EXAMPLE 2 Evaluate this third-order determinant:

.

We calculate as follows:

.


Solving Systems Using Determinants

Here is a system of two equations in two variables:

,

.

We form three determinants, which we call D, , and .

In D, we have the coefficients of x and y.

To form , we replace the x-coefficients in Dwith the constants on the right side of theequations.

To form , we replace the y-coefficients in Dwith the constants on the right.

It is important that the replacement be done without changing the order of thecolumns. Then the solution of the system can be found as follows. This isknown as Cramer’s rule.

Dy Dy �a1

a2

c1

c2

Dx

Dx �c1

c2

b1

b2

D � �a1

a2

b1

b2�

DyDx

a2x � b2 y � c2

a1x � b1 y � c1

� �53

� �9 � 40 � 4

� �1�9� � 5��8� � 4��1� � �1�1 � 1 � 8��1�� 5�0 � 1 � 8 � 1� � 4�0 � ��1� � 1 � 1�

�1�18 �11� � ��5��08 1

1� � 4�01 1�1�

�1

�5

4

0

1

8

1

�1

1

� �1 �18 �11� � � �5 � �08 1

1� � 4 �01 1�1�

a1

a2

a3

b1

b2

b3

c1

c2

c3

a3

a1

a2

a3

b1

b2

b3

c1

c2

c3

a2

a1

a2

a3

b1

b2

b3

c1

c2

c3

a1

Evaluate.

3.

4.


� 3�2

4

21

�3

243�

�213

�124

1�1�3

�

980


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CRAMER’S RULE

,

EXAMPLE 3 Solve using Cramer’s rule:

,

.

We compute D, , and :

;

;

.

Then

, or and .

The solution is .

Do Exercise 5.

Cramer’s rule for three equations is very similar to that for two.

D is again the determinant of the coefficients of x, y, and z. This time we haveone more determinant, . We get it by replacing the z-coefficients in D withthe constants on the right:

.Dz �

a1

a2

a3

b1

b2

b3

d1

d2

d3

Dz

Dy �

a1

a2

a3

d1

d2

d3

c1

c2

c3

Dx �

d1

d2

d3

b1

b2

b3

c1

c2

c3

D � �a1

a2

a3

b1

b2

b3

c1

c2

c3�

a3x � b3 y � c3z � d3

a2x � b2 y � c2z � d2

a1x � b1 y � c1z � d1

�83 , 1

2�

y �Dy

D�

612

�12

83

x �Dx

D�

3212

Dy �33

79

� 3 � 9 � 3 � 7 � 27 � 21 � 6

Dx �79

�22

� 7 � 2 � 9��2� � 14 � 18 � 32

D � �33 �22� � 3 � 2 � 3 � ��2� � 6 � 6 � 12

DyDx

3x � 2y � 9

3x � 2y � 7

y �Dy

Dx �

Dx

D

5. Solve using Cramer’s rule:

,

.

Answer on page A-63

x � 3y � 0

4x � 3y � 15

981


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The solution of the system is given by

CRAMER’S RULE

, ,

EXAMPLE 4 Solve using Cramer’s rule:

,

,

.

We compute D, , , and :

; ;

; .

Then

;

;

.

The solution is .

In Example 4, we would not have needed to evaluate . Once we foundx and y, we could have substituted them into one of the equations to find z.In practice, it is faster to use determinants to find only two of the numbers;then we find the third by substitution into an equation.

Do Exercise 6.

In using Cramer’s rule, we divide by D. If D were 0, we could not do so.

INCONSISTENT SYSTEMS;DEPENDENT EQUATIONS

If and at least one of the other determinants is not 0, then thesystem does not have a solution, and we say that it is inconsistent.

If and all the other determinants are also 0, then there is aninfinite set of solutions. In that case, we say that the equations in the system are dependent.

D � 0

D � 0

Dz

��2, 35 , 12

5 �

z �Dz

D�

�24�10

�125

y �Dy

D�

�6�10

�35

x �Dx

D�

20�10

� �2

Dz �

111

�31

�2

1314

� �24Dy �

111

1314

713

� �6

Dx �

1314

�31

�2

713

� 20D � �111

�31

�2

713� � �10

DzDyDx

x � 2y � 3z � 4

x � y � z � 1

x � 3y � 7z � 13

z �Dz

Dy �

Dy

Dx �

Dx

D

6. Solve using Cramer’s rule:

Answer on page A-63

4x � y � 7.

2x � 3y � z � 9,

x � 3y � 7z � 6,

982


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983

Exercise Set I

EXERCISE SETIIEvaluate.

1. 2. 3. 4. � 4�7

59��3

�5�6

�10��54 4�5��32 7

8�

5. 6. 7. 8. �00 �4�6��20 �7

0��19 18�� 8

122

�3�

Evaluate.

9. 10. 11. 12. �123

213

201��1

30

�241

�322��35

2

010

�22

�1��03

1

2�1�2

012�

13. 14. 15. 16. ��125

643

�54

10��31

1

211

411��21

3

�124

1�1�3

�� 3�2�4

21

�3

�243�

Solve using Cramer’s rule.

17. , 18. , 19. , 20. ,7x � 2y � 65x � 4y � �3

3x � 7y � 1�2x � 4y � 3

3x � 7y � 55x � 8y � 1

5x � 9y � 103x � 4y � 6

21. , 22. , 23. , 24. ,�3x � 2y � 13

x � 4y � 53x � 5y � 3

x � 4y � 89x � 2y � 53x � 3y � 11

3x � y � 24x � 2y � 11

25. ,,

26. ,,

27. ,,

28. ,,

7a � 3b � 5c � 14b � 2c � 2

a � 3c � 6

r � s � t � 62r � s � t � �3

r � 2s � 3t � 6

2x � y � z � �3x � 2y � 3z � 4x � y � 2z � �3

5x � y � 4z � 27x � 2y � z � �4

2x � 3y � 5z � 27

29. ,,

30. ,,

31. ,,

32. ,,

3x � y � 4z � 32x � y � 2z � �8

x � 2y � 3z � 9

4p � 5q � 6r � 4p � 2q � 3r � 3p � q � r � 1

2x � 4y � z � 0x � 2y � z � 5

3x � 2y � 2z � 3

2x � y � 2z � 58x � y � z � 54x � y � 3z � 1

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984

APPENDIX J: Elimination Using Matrices

The elimination method concerns itself primarily with the coefficientsand constants of the equations. In what follows, we learn a method for solv-ing systems using just the coefficients and the constants. This procedure in-volves what are called matrices.

In solving systems of equations, we perform computations with theconstants. The variables play no important role until the end. Thus we cansimplify writing a system by omitting the variables. For example, the system

,

x � 2y � 1

3x � 4y � 5

JJ ELIMINATION USING MATRICESObjectiveSolve systems of two or threeequations using matrices.

simplifies to

if we omit the variables, the operation of addition, and the equals signs. Theresult is a rectangular array of numbers. Such an array is called a matrix(plural, matrices). We ordinarily write brackets around matrices. The follow-ing are matrices.

The rows of a matrix are horizontal, and the columns are vertical.

column 1 column 2 column 3

Let’s now use matrices to solve systems of linear equations.

EXAMPLE 1 Solve the system

We write a matrix using only the coefficients and the constants, keepingin mind that x corresponds to the first column and y to the second. A dashedline separates the coefficients from the constants at the end of each equation:

The individual numbers are called elements or entries.

Our goal is to transform this matrix into one of the form

The variables can then be reinserted to form equations from which we cancomplete the solution.

�a0

bd

ce�.

� 5�2

�43

�12�.

�2x � 3y � 2.

5x � 4y � �1,

row 1row 2row 3

212��2

01

�510

�416

103

31

�2

520�, �6

14

220

11

�2

430

71

�3�,

1145�7

80

20910

.

3 �4 5

1 �2 1

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We do calculations that are similar to those that we would do if we wrotethe entire equations. The first step, if possible, is to multiply and/or inter-change the rows so that each number in the first column below the first num-ber is a multiple of that number. In this case, we do so by multiplying Row 2by 5. This corresponds to multiplying the second equation by 5.

New Row (Row 2)

Next, we multiply the first row by 2 and add the result to the second row. Thiscorresponds to multiplying the first equation by 2 and adding the result to thesecond equation. Although we write the calculations out here, we generallytry to do them mentally:

New Row

If we now reinsert the variables, we have

(1)

(2)

We can now proceed as before, solving equation (2) for y:

(2)

Next, we substitute for y back in equation (1). This procedure is called back-substitution.

(1)

Substituting for y in equation (1)

Solving for x

The solution is

Do Exercise 1.

EXAMPLE 2 Solve the system

We first write a matrix, using only the coefficients and the constants.Where there are missing terms, we must write 0’s:

.

Our goal is to find an equivalent matrix of the form

A matrix of this form can be rewritten as a system of equations from which asolution can be found easily.

�a00

be0

cf

h

dgi�.

(P1), (P2), and (P3) designate theequations that are in the first,second, and third position,respectively.

(P1)(P2)(P3)

�216

�10

�1

4�4

2

�35

10�

6x � y � 2z � 10.

x � 4z � 5,

2x � y � 4z � �3,

�57 , 8

7�. x � 5

7

87 5x � 4 �

87 � �1

5x � 4y � �1

87

y � 87 .

7y � 8

7y � 8.

5x � 4y � �1,

2 � 2�Row 1� � �Row 2��50

�47

�18�

2 � 5 � ��10� � 0; 2��4� � 15 � 7; 2��1� � 10 � 8.

2 � 5� 5�10

�415

�110�

1. Solve using matrices:

Answer on page A-63

2x � 5y � �6.

5x � 2y � �44,

985


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The first step, if possible, is to interchange the rows so that each numberin the first column below the first number is a multiple of that number. In thiscase, we do so by interchanging Rows 1 and 2:

.

Next, we multiply the first row by �2 and add it to the second row:

Now we multiply the first row by �6 and add it to the third row:

Next, we multiply Row 2 by �1 and add it to the third row:

Reinserting the variables gives us

(P1)

(P2)

(P3)

We now solve (P3) for z:

(P3)

Solving for z

Next, we back-substitute for z in (P2) and solve for y :

(P2)

Substituting for z in equation (P2)

Solving for y

Since there is no y-term in (P1), we need only substitute for z in (P1) andsolve for x:

(P1)

Substituting for z in equation (P1)

Solving for x

The solution is

Do Exercise 2.

�3, 7, �12�.

x � 3.

x � 2 � 5

�12 x � 4��

12� � 5

x � 4z � 5

�12

y � 7.

�y � �7

�y � 6 � �13

�12 �y � 12��

12� � �13

�y � 12z � �13

�12

z � �12 .

z � �7

14

14z � �7

14z � �7.

�y � 12z � �13,

x � 4z � 5,

This corresponds to multiplying equation(P2) by �1 and adding it to equation (P3).�1

00

0�1

0

�41214

5�13

�7�.

This corresponds to multiplying equation(P1) by �6 and adding it to equation (P3).�1

00

0�1�1

�41226

5�13�20

�.

This corresponds to multiplying newequation (P1) by �2 and adding it to newequation (P2). The result replaces theformer (P2). We perform the calculationsmentally.

�106

0�1�1

�412

2

5�13

10�.

This corresponds to interchangingthe first two equations.�1

26

0�1�1

�442

5�310�

2. Solve using matrices:

Answer on page A-63

4x � y � z � 1.

2x � y � z � �1,

x � 2y � 3z � 4,

986


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All the operations used in the preceding example correspond to opera-tions with the equations and produce equivalent systems of equations. Wecall the matrices row-equivalent and the operations that produce them row-equivalent operations.

ROW-EQUIVALENT OPERATIONS

Each of the following row-equivalent operations produces anequivalent matrix:

a) Interchanging any two rows.b) Multiplying each element of a row by the same nonzero number.c) Multiplying each element of a row by a nonzero number and

adding the result to another row.

The best overall method of solving systems of equations is by row-equivalent matrices; graphing calculators and computers are programmedto use them. Matrices are part of a branch of mathematics known as linearalgebra. They are also studied in more detail in many courses in finitemathematics.

987


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988


Co

pyr

igh

t ©

200

7 P

ears

on

Ed

uca

tio

n, I

nc.

JJ EXERCISE SET

Solve using matrices.

1. 2. 3.3x � 5y � 3

x � 4y � 8,9x � 2y � 53x � 3y � 11,

3x � y � 24x � 2y � 11,

4. 5. 6.�5x � 2y � 10

3x � 4y � 7,4x � 2y � 55x � 3y � �2,

�3x � 2y � 13x � 4y � 5,

7. 8. 9.

2x � y � 2z � 58x � y � z � 5,4x � y � 3z � 1,

7x � 9y � 114x � 5y � �8,

5x � y � 402x � 3y � 50,

10. 11. 12.

3x � y � 4z � 32x � y � 2z � �8,

x � 2y � 3z � 9,

4p � 5q � 6r � 4p � 2q � 3r � 3,p � q � r � 1,

2x � 4y � z � 0x � 2y � z � 5,

3x � 2y � 2z � 3,

13. 14. 15.

p � 6q � 1q � 7r � 4,3p � 2r � 11,

6b � 6c � �18a � 6c � �1,4a � 9b � 8,

2x � 2y � z � �3x � 2y � 3z � �1,x � y � 2z � 0,

16. 17. , 18.

x � y � z � w � �14x � y � z � w � 22,x � y � z � w � �4,

2x � 3y � z � w � �8,

3x � 2y � 2z � w � �6x � y � 4z � 3w � �2,x � y � z � w � �5,

2x � 2y � 2z � 2w � �10

m � 2n � t � 5m � n � t � 0,m � n � t � 6,

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989

APPENDIX K: The Algebra of Functions

The Sum, Difference, Product, and Quotient of Functions

Suppose that a is in the domain of two functions, f and g. The input a is pairedwith by f and with by g. The outputs can then be added to get

EXAMPLE 1 Let and Find

We visualize two function machines. Because 2 is in the domain of eachfunction, we can compute and

Since

we have

In Example 1, suppose that we were to write as or This could then be regarded as a “new”

function: We can alternatively find with

Substituting 2 for x

Similar notations exist for subtraction, multiplication, and division offunctions.

THE SUM, DIFFERENCE, PRODUCT,AND QUOTIENT OF FUNCTIONS

For any functions f and g, we can form new functions defined as:

1. The sum2. The difference3. The product fg :4. The quotient � f�g� �x� � f �x��g�x�, where g�x� � 0.f�g :

� f � g� �x� � f �x� � g�x�;� f � g� �x� � f �x� � g�x�;f � g :� f � g� �x� � f �x� � g�x�;f � g :

� 11.

� 4 � 2 � 5

� f � g� �2� � 22 � 2 � 5

� f � g� �x� � x2 � x � 5

� f � g� �x�:f �2� � g�2�� f � g� �x� � x2 � x � 5.

f �x� � g�x� � x2 � x � 5.�x2 � 1�,�x � 4� �f �x� � g�x�

f �2� � g�2� � 6 � 5 � 11.

f �2� � 2 � 4 � 6 and g�2� � 22 � 1 � 5,

g(x) � x 2 � 1

f(x) � x � 4

f(2)

g(2)

f

g

2

2

g�2�.f �2�

f �2� � g�2�.g�x� � x2 � 1.f �x� � x � 4

f �a� � g�a�.g�a�f �a�

KK THE ALGEBRA OF FUNCTIONSObjectiveGiven two functions fand g, find their sum,difference, product, and quotient.

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EXAMPLE 2 Given f and g described by and find and

Note that the sum, difference, and product of polynomials are also poly-nomial functions, but the quotient may not be.

Do Exercise 1.

EXAMPLE 3 For and find and

We first find and

Then we substitute.

Using

Using

Using

Using

Do Exercise 2.

�16 � 4

�2�

20�2

� �10

� f�g� �x� � �x2 � x��x � 2� � f�g� ��4� ��4�2 � ��4�

�4 � 2

� 125 � 25 � 10 � 140;

� f � g� �x� � x3 � x2 � 2x � f � g� �5� � 53 � 52 � 2 � 5

� 1 � 2 � 2 � 1;

� f � g� �x� � x2 � 2x � 2 � f � g� ��1� � ��1�2 � 2��1� � 2

� 9 � 2 � 11;

� f � g� �x� � x2 � 2 � f � g� �3� � 32 � 2

� f�g� �x� �f �x�g�x�

�x2 � xx � 2

.

� x3 � x2 � 2x;

� x3 � 2x2 � x2 � 2x

� f � g� �x� � f �x� � g�x� � �x2 � x� �x � 2�

� x2 � 2x � 2;

� x2 � x � x � 2

� f � g� �x� � f �x� � g�x� � x2 � x � �x � 2�

� x2 � 2;

� f � g� �x� � f �x� � g�x� � x2 � x � x � 2

� f�g� �x�.� f � g� �x�,� f � g� �x�,� f � g� �x�,� f�g� ��4�.� f � g� �5�,� f � g� ��1�,

� f � g� �3�,g�x� � x � 2,f �x� � x2 � x

�g � g� �x� � g�x� � g�x� � �x � 7� �x � 7� � x2 � 14x � 49

� f�g� �x� � f �x��g�x� �x2 � 5x � 7

;

� f � g� �x� � f �x� � g�x� � �x2 � 5� �x � 7� � x3 � 7x2 � 5x � 35;

� f � g� �x� � f �x� � g�x� � �x2 � 5� � �x � 7� � x2 � x � 12;

� f � g� �x� � f �x� � g�x� � �x2 � 5� � �x � 7� � x2 � x � 2;

�g � g� �x�.� f�g� �x�,� f � g� �x�,� f � g� �x�,� f � g� �x�,g�x� � x � 7,f �x� � x2 � 51. Given and

find each of thefollowing.

a)

b)

c)

d)

e)

2. Given andfind each of the

following.

a)

b)

c)

d)


� f�g� �2�

� f � g� ��3�

� f � g� �4�

� f � g� ��2�

g�x� � 2x � 3,f �x� � x2 � x

� f � f � �x�

� f�g� �x�

� f � g� �x�

� f � g� �x�

� f � g� �x�

g�x� � x2 � 3,f �x� � x2 � 3

990

APPENDIX K: The Algebra of Functions

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991

Exercise Set K

EXERCISE SETKK

1. 2. 3. f �5� � g�5�f ��1� � g��1�f �2� � g�2�

4. 5. 6. f ��2� � g��2�f ��1� � g��1�f �4� � g�4�

7. 8. 9. g�1� � f �1�f �3��g�3�f ��4��g��4�

10. 11. 12. g�6� � f �6�g�0��f �0�g�2��f �2�

Let and Find the following.g�x� � 4 � x.f �x� � x2 � 3

13. 14. 15. � f � g� ��4�� f � g� �x�� f � g� �x�

16. 17. 18. � f � g� �2�� f � g� �3�� f � g� ��5�

19. 20. 21. � f � g� ��3�� f�g� �x�� f � g� �x�

22. 23. 24. � f�g� �1�� f�g� �0�� f � g� ��4�

25. 26. � f�g� ��1�� f�g� ��2�

For each pair of functions f and g, find and � f�g� �x�.� f � g� �x�,� f � g� �x�,� f � g� �x�,

27. , 28. 29.

g�x� � 4x3

f �x� �1

x � 2,

g�x� � 2x2f �x� � 5x � 1,

g�x� � 3x � 4f �x� � x2

30. 31. 32.

g�x� �1

x � 2

f �x� �5

x � 3,

g�x� �5

4 � x

f �x� �3

x � 2,

g�x� �1

x � 4

f �x� � 3x2,

Let and Find the following.g�x� � x2 � 2.f �x� � �3x � 1

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992

APPENDIX L: Distance, Midpoints, and Circles

In carpentry, surveying, engineering, and other fields, it is often neces-sary to determine distances and midpoints and to produce accurately drawn circles.

The Distance Formula

Suppose that two points are on a horizontal line, and thus have the same sec-ond coordinate. We can find the distance between them by subtracting theirfirst coordinates. This difference may be negative, depending on the order inwhich we subtract. So, to make sure we get a positive number, we take the ab-solute value of this difference. The distance between two points on a horizon-tal line and is thus Similarly, the distance betweentwo points on a vertical line and is

Now consider any two points and If and these points are vertices of a right triangle, as shown. The other vertex is then

The lengths of the legs are and We find d, thelength of the hypotenuse, by using the Pythagorean theorem:

Since the square of a number is the same as the square of its opposite, wedon’t need these absolute-value signs. Thus,

Taking the principal square root, we obtain the distance between two points.

THE DISTANCE FORMULA

The distance between any two points and is given by

This formula holds even when the two points are on a vertical or a hori-zontal line.

d � ��x2 � x1�2 � � y2 � y1�2.

�x2, y2��x1, y1�

d 2 � �x2 � x1�2 � � y2 � y1�2.

d 2 � �x2 � x1�2 � �y2 � y1�2.

�y2 � y1�.�x2 � x1��x2, y1�.

y1 � y2,x1 � x2�x2, y2�.�x1, y1�

�y2 � y1�

�x2 � x1�(x1, y1) (x2, y1)

(x2, y2)

d

(x1, y1) (x2, y1)

The distance betweenx1 and x2 � �x2 � x1�

x

y

�y2 � y1�.�x2, y2��x2, y1��x2 � x1�.�x2, y1��x1, y1�

LL DISTANCE, MIDPOINTS, AND CIRCLES

ObjectivesUse the distance formula tofind the distance betweentwo points whosecoordinates are known.

Use the midpoint formula tofind the midpoint of asegment when thecoordinates of its endpointsare known.

Given an equation of a circle,find its center and radiusand graph it; and given thecenter and radius of a circle,write an equation of thecircle and graph the circle.

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EXAMPLE 1 Find the distance between and Give an exactanswer and an approximation to three decimal places.

We substitute into the distance formula:

Substituting

Using a calculator


Midpoints of Segments

The distance formula can be used to derive a formula for finding the midpointof a segment when the coordinates of the endpoints are known.

THE MIDPOINT FORMULA

If the endpoints of a segment are and then thecoordinates of the midpoints are

(To locate the midpoint, determinethe average of the x-coordinatesand the average of they-coordinates.)

EXAMPLE 2 Find the midpoint of the segment with endpoints and

Using the midpoint formula, we obtain


��2 � 42

,3 � ��6�

2 �, or � 22

,�32 �, or �1, �

32 �.

�4, �6�.��2, 3�

�x1 � x2

2,

y1 � y2

2 �.

�x2, y2�,�x1, y1�

�4 �2 4

�4

�2

2

4

�5 �3 �1 3 5

�5

�3

�1

1

3

5

(4, �3)

(�5, 4)

�130 � 11.402

x

y

� �130 � 11.402.

� ��9�2 � 72

d � ��5 � 4�2 � 4 � ��3�2

��5, 4�.�4, �3� Find the distance between the pairof points. Where appropriate, givean approximation to three decimalplaces.

1. and

2. and

Find the midpoint of the segmentwith the given endpoints.

3. and

4. and


�8, �3��10, �7�

�6, �7��3, 1�

�4, 2��2, 1�

��4, �2��2, 6�

993


(x2, y2)

(x1, y1)

x

y

x1 � x2

2y1 � y2

2,� �

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Circles

Another conic section, or curve, shown in the figure at the beginning of thissection is a circle. A circle is defined as the set of all points in a plane that area fixed distance from a point in that plane.

Let’s find an equation for a circle. We call the center and let the ra-dius have length r. Suppose that is any point on the circle. By the dis-tance formula, we have

Squaring both sides gives an equation of the circle in standard form:When and the circle is centered at the

origin. Otherwise, we can think of that circle being translated units hori-zontally and units vertically from the origin.

EQUATIONS OF CIRCLES

A circle centered at the origin with radius r has equation

A circle with center and radius r has equation

(Standard form)

EXAMPLE 3 Find the center and the radius and graph this circle:

First, we find an equivalent equation in standard form:

Thus the center is and theradius is 4. We draw the graph,shown at right, by locating thecenter and then using a compass,setting its radius at 4, to drawthe circle.


x

y

�2 2

�2

2

4

�5�6�7 �3 �1 3�1

3

5

6

7

8

1

1

(x � 2)2 � (y � 3)2 � 16

(�2, 3)

��2, 3�

x � ��2�2 � � y � 3�2 � 42.

�x � 2�2 � � y � 3�2 � 16.

�x � h�2 � � y � k�2 � r 2.

�h, k�

x 2 � y 2 � r 2.

k

h(0, 0)

(h, k)

(x, y)r

(x, y)r

x

y

x 2 � y 2 � r 2

(x � h)2 � (y � k)2 � r 2

�k��h�

k � 0,h � 0�x � h�2 � � y � k�2 � r 2.

��x � h�2 � � y � k�2 � r.

�x, y��h, k�

5. Find the center and the radius of the circle

Then graph the circle.

6. Find the center and the radius of the circle


x2 � y2 � 64.

�8 �4 4 8

�8

�4

4

8

�10�12 �6 �2 2 6 10 12

�10

�12

�6

�2

2

6

10

12

x

y

�x � 5�2 � � y �12�2 � 9.

994


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EXAMPLE 4 Write an equation of a circle with center and radius

We use standard form and substitute:

Substituting

Simplifying

Do Exercise 7.

With certain equations not in standard form, we can complete the squareto show that the equations are equations of circles. We proceed in much thesame manner as we did in Section 11.6.

EXAMPLE 5 Find the center and the radius and graph this circle:

First, we regroup the terms and then complete the square twice, oncewith and once with

Regrouping

Adding 0

substitutingand

Regrouping

Factoring andsimplifying

Adding 2

Writingstandardform

The center is and the radius is

Do Exercise 8.

�2

2

4

�5�6 �3 �1�1

3

5

1

1

x 2 � y 2 � 8x � 2y � 15 � 0

(�4, 1)

[x � (�4)]2 � (y � 1)2 � (œ2)2Í

x

y

�2.��4, 1�

x � ��4�2 � � y � 1�2 � ��2 �2.

�x � 4�2 � � y � 1�2 � 2

�x � 4�2 � � y � 1�2 � 2 � 0

�x 2 � 8x � 16� � � y 2 � 2y � 1� � 16 � 1 � 15 � 0

1 � 116 � 16

��22 �2 � 1;

�82�2 � 42 � 16; �x 2 � 8x � 16 � 16� � � y 2 � 2y � 1 � 1� � 15 � 0

�x 2 � 8x � 0� � � y 2 � 2y � 0� � 15 � 0

�x 2 � 8x� � � y 2 � 2y� � 15 � 0

x 2 � y 2 � 8x � 2y � 15 � 0

y 2 � 2y :x 2 � 8x

x 2 � y 2 � 8x � 2y � 15 � 0.

�x � 9�2 � � y � 5�2 � 2.

�x � 9�2 � y � ��5�2 � ��2 �2

�x � h�2 � � y � k�2 � r 2

�2.�9, �5� 7. Find an equation of a circle with

center and radius 6.

8. Find the center and the radiusof the circle

Then graph the circle.


x

y

�4 �2 2 4

�4

�2

2

4

�5 �3 �1 3 5

�5

�3

�1

3

5

1

1

x2 � 2x � y2 � 4y � 2 � 0.

��3, 1�

995


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996


Find the distance between the pair of points. Where appropriate, give an approximation to three decimal places.

Co

pyr

igh

t ©

200

7 P

ears

on

Ed

uca

tio

n, I

nc.

LL EXERCISE SET

1. and 2. and 3. and �5, �6��0, �4��4, 14��1, 2��2, �7��6, �4�

4. and 5. and 6. and ��8, 1��2, 22��9, �9��9, 9��8, �3��8, 3�

7. and 8. and 9. and � 17

,1114�� 5

7,

114��5.6, �4.4��6.1, 2��4.3, �3.5��2.8, �3.5�

10. and 11. and 12. and ��46, �38��34, �18��56, �17��23, 10��6, 0��0, �7 �

13. and 14. and 15. and ��7, �5 ��2, ��3 ��p, q��0, 0��0, 0��a, b�

16. and 17. and 18. and ��430, �640��3000, 560��2000, 580��1000, �240��5, ��6 ��8,�3 �

19. and 20. and 21. and ��3, 6��3, 5��2, �4��5, 10��4, �2��1, 9�

22. and 23. and 24. and ��5, 12��6, �2��8, �4��10, �13��4, 11��7, �3�

25. and 26. and 27. and ��13

,56�� 1

6, �

34��5.2, �6.9��4.1, 6.9��2.9, �8.7��3.4, 8.1�

Find the midpoint of the segment with the given endpoints.

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997

Exercise Set L

28. and 29. and 30. and ��4, 5�3 ��9, 2�3 ��3, 4��2, �1�� 18

,34��

45

, �23�

Find the center and the radius of the circle. Then graph the circle.

31. 32. 33.

�4 �2 2 4

�4

�2

2

4

�5 �3 �1 1 3 5

�5

�3

�1

1

3

5

x

y

�x � 3�2 � y2 � 2

�4 �2 2 4

�4

�2

2

4

�5 �3 �1 1 3 5

�5

�3

�1

1

3

5

x

y

�x � 2�2 � � y � 3�2 � 1

�4 �2 2 4

�4

�2

2

4

�5 �3 �1 1 3 5

�5

�3

�1

1

3

5

x

y

�x � 1�2 � � y � 3�2 � 4

34. 35. 36.

�4 �2 2 4

�4

�2

2

4

�5 �3 �1 1 3 5

�5

�3

�1

1

3

5

x

y

x2 � y2 � 9

�4 �2 2 4

�4

�2

2

4

�5 �3 �1 1 3 5

�5

�3

�1

1

3

5

x

y

x2 � y2 � 25

�4 �2 2 4

�4

�2

2

4

�5 �3 �1 1 3 5

�5

�3

�1

1

3

5

x

y

x2 � � y � 1�2 � 3

Find an equation of the circle having the given center and radius.

37. Center radius 7 38. Center radius 4�0, 0�,�0, 0�,

39. Center radius 40. Center radius 3�2�4, 1�,�7��5, 3�,

Find the center and the radius of the circle.

41. 42. 43. x2 � y2 � 8x � 2y � 13 � 0x2 � y2 � 6x � 4y � 15 � 0x2 � y2 � 8x � 6y � 15 � 0

44. 45. 46. x2 � y2 � 10y � 75 � 0x2 � y2 � 4x � 0x2 � y2 � 6x � 4y � 12 � 0

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Photo Credits

4, © Rob Howard/CORBIS 11, AP Wire Services 19, © Buddy Mays/CORBIS28, © Douglas Peebles/CORBIS 77, Comstock Images 117, © Jim Craigmyle/CORBIS 120 (top), PhotoDisc 120 (bottom), Brian Spurlock 122, AppalachianTrail Conference 125, Corbis 128, © Patrik Giardino/CORBIS 131, Doug Meneuz,PhotoDisc 134, EyeWire 135, © Ed Bock/CORBIS 137, EyeWire 152, © JeremyBirch 156, © Charles O’Rear/CORBIS 158, Andy Sacks, Stone/Getty 162, ©Lawrence Migdale, Tony Stone 201, Karen Bittinger 207 (left), Mary Clay/TomStack and Associates 207 (right), Terje Rakke/The Image Bank 213, Andy Sacks,Stone/Getty 235 (left), © Dave G. Houser/CORBIS 235 (middle), PhotoDisc235 (right), EyeWire 239 (left), © Digital Art/CORBIS 239 (right), PhotoDisc242 (right), Hoby Finn, PhotoDisc 243, Bob Daemmrich/The Image Works254, Ron Chapple, FPG International 286, Brian Spurlock 290, © Nathan Benn/CORBIS 341, PhotoDisc 377, © LE SEGRETAIN PASCAL/CORBIS SYGMA380, PhotoDisc Collection 383 (top), Duncan Smith/PhotoDisc 383 (bottom),EyeWire 446, © Kelly-Mooney Photography/CORBIS 451 (left), © 1990 GlennRandall 451 (right), PhotoDisc 452, © Anne Ryan/New Sport/Corbis457, PhotoLine/PhotoDisc 458 (left), Geostock/PhotoDisc 458 (right), TomTurpin, Purdue University 459 (left), Reuters/Anthony P. Bolante/ReutersNewMedia Inc./CORBIS 459 (right), © Reuters/CORBIS 460, Joseph Sohm,ChromoSohn 464, © Royalty-Free/Corbis 465, PhotoDisc 468, EyeWireCollection 469, Digital Vision/Getty 470, Stockbyte/Getty 474, Hisham E.Ibrahim, PhotoDisc 475, Steve Mason, PhotoDisc 478 (left), © Richard Cummins/CORBIS 478 (right), AFP Photo/Jeff Haynes, CORBIS 489, © Royalty-Free/Corbis492, Phil Schermeister, Corbis 509, Karen Bittinger 525, 532, 533, © Royalty-Free/Corbis 538, BrandXPictures/Getty 548, EyeWire Collection 559, Angelo Hornak/Corbis 573, Susan Dawson 584, © Tom Stewart/CORBIS 597, © Viviane Moos/Corbis 598, Stephanie Maze/Corbis 599, © Grace/zefa/Corbis 600, KarenBittinger 610, Dave G. Houser/Corbis 621, © David Butow/CORBIS SABA622, PhotoDisc 630, Keith Brofsky/PhotoDisc 640, © Charles O’Rear/CORBIS668, Warren Morgan/Corbis 671, © Duomo/Corbis 674, © LE SEGRETAINPASCAL/CORBIS SYGMA 686, © Royalty-Free/Corbis 723, © 1999 Bill Ellzey727, © Galen Rowell/CORBIS 728, EyeWire Collection 763, AP/Photo766 (left), Corbis 766 (right), 776, PhotoDisc 858 (top), © Royalty-Free/Corbis858 (bottom), Digital Vision 869, © Michael S. Yamashita/CORBIS 911, SarahLawless/Stone/Getty Images 913, © Royalty-Free/Corbis 914, Alan ScheinPhotography/Corbis 916, John Lacko 917, 921, © Royalty-Free/Corbis922, © B. Kohlhas/zefa/Corbis 924 (left), From Classic Baseball Cards, by BertRandolph Sugar, copyright © 1977 by Dover Publishing, Inc. 924 (right), © FrancisG. Mayer/Corbis 925, © Royalty-Free/Corbis 928, Stockbyte/Getty Images929, © Franco Vogt/CORBIS 934, PhotoDisc 973, © Viviane Moos/Corbis975, © Corbis Images

998

PHOTO CREDITS

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A-1

Chapter 1

CHAPTER 1

Margin Exercises, Section 1.1, pp. 2–6

1. 890 ft 2. 64 3. 28 4. 605. 192 6. 25 7. 16 8. 12 hr 9.

10. or 11. 12.

13. or 14.15. or 16. 17.

Calculator Corner, p. 4

1. 56 2. 11.9 3. 1.8 4. 34,427.16 5. 20.16. 29.9

Exercise Set 1.1, p. 7

1. 32 min; 69 min; 81 min 3. 1935 5. 260 mi7. 24 9. 56 11. 8 13. 1 15. 6 17. 219. or 21. 23. , or

25. or 27. or or or

29. or 31. 33. or 35. 2z 37. 3m 39. or 41.43. 45. or 47. or 49. or 0.89s, where s is the salary 51.

53. 65t miles 55. 57. 59. 61. 0


1. 8; �5 2. 125; �50 3. �3 4. �10; 1565. �120; 50; �80 6.

7.

8. 9. �0.375

0

�114

1�1�2�3�4�5�6 2 3 4 65

0 1�1�2�3�4�5�6 2 3 4 65

�1.4

0

��72

1�1�2�3�4�5�6 2 3 4 65

14

DW$50 � x

s � 0.05s89%s,3y � 4x4x � 3y,

11 � 3n3n � 11,2t � 5xy � 86 � 4a4a � 6,

y � xx � y,n � mw � xx � w,

x �1y

x�y,xy

,x � y,b � aa � b,

4 � qq � 4c � 127 � bb � 7,ft2

m2

p � qxy � 2000.59x59%x,a � b8x � 66 � 8x,

12

pm � 48 � yy � 8,

x � 8ft214,410 � x � 15,300;

10. 11. 12. 13. 14. 15.16. 17. 18. 19. 20.21. 22. False 23. True 24. True 25. 8

26. 9 27. 28. 5.6


1. �0.75 2. �0.45 3. �0.125 4. �1.8 5. �0.6756. �0.6875 7. �3.5 8. �0.76


1. 8.717797887 2. 17.80449381 3. 67.082039324. 35.4807407 5. 3.141592654 6. 91.106186957. 530.9291585 8. 138.8663978


1. 5 2. 17 3. 0 4. 6.48 5. 12.7 6. 0.9

7. 8.


1. �34,000,000 3. 24; �2 5. 950,000,000; �4607. Alley Cats: �34; Strikers: 349.

11.

13. 15. �0.875

17. 19. 21. 23. 0.1 25. �0.527. 0.16 29. 31. 33. 35. 37.39. 41. 43. 45. 47. 49.51. 53. True 55. False 57.59. 61. 3 63. 10 65. 0 67. 30.4y � �10

x � �6��

��0.6�1.160.83

�4�25

0 1�1�2�3�4�6 �5 2 3 4 5 6

0

�5.2

1�1�2�3�4�5�6 2 3 4 5 6

0

�103

1�1�2�3�4�5�6 2 3 4 65

43

57

23

4 � x7 � �5��

��1.3�0.54

Answers

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69. 71. 0 73. 75. 77. 3 78. 11

79. 3 80. 1 81.

83. �8.76, �5.16, �4.24, �2.13, 1.85, 5.23

85. 87. or


1. �3 2. �3 3. �5 4. 4 5. 0 6. �2 7. �118. �12 9. 2 10. �4 11. �2 12. 0 13. �2214. 3 15. 0.53 16. 2.3 17. �7.7 18. �6.2

19. 20. 21. �58 22. �56 23. �14

24. �12 25. 4 26. �8.7 27. 7.74 28. 29. 0

30. �12 31. �14; 14 32. �1; 1 33. 19; �19

34. 1.6; �1.6 35. 36. 37. 4

38. 13.4 39. 0 40. 41. �2 students


1. �7 3. �6 5. 0 7. �8 9. �7 11. �2713. 0 15. �42 17. 0 19. 0 21. 3 23. �925. 7 27. 0 29. 35 31. �3.8 33. �8.1

35. 37. 39. 41. 43.

45. 47. 49. 37 51. 50 53. �1409

55. �24 57. 26.9 59. �8 61. 63. �43

65. 67. 24 69. 71. 13,796 ft 73. �3�F

75. �$20,300 77. He owes $85. 79. 81. �0.62582. 83. 84. 0.65 85. 2.3 86. 0

87. 88. 21.4 89. All positive 91. (b)


1. �10 2. 3 3. �5 4. �1 5. 2 6. �4 7. �28. �11 9. 4 10. �2 11. �6 12. �16 13. 7.1

14. 3 15. 0 16. 17. �8 18. 7 19. �3

20. �23.3 21. 0 22. �9 23. 24. 12.7

25. 214�F higher


1. �7 3. �6 5. 0 7. �4 9. �7 11. �613. 0 15. 14 17. 11 19. �14 21. 5 23. �1

�

65

32

45

�0.0830.3

DW

38

43

138

1645

815

124

�1924

�38

�79

�15

�14

�98

98

;23

�23

;

89

�1920

�29

509

559

,19

12

38

,16

,�23

,�34

,�56

,

DW358

23

A-2

Answers

25. 18 27. �3 29. �21 31. 5 33. �8 35. 1237. �23 39. �68 41. �73 43. 116 45. 0

47. �1 49. 51. 53. 55. 19.9

57. �8.6 59. �0.01 61. �193 63. 500

65. �2.8 67. �3.53 69. 71. 73.

75. 77. 79. 81. 37 83. �62

85. �139 87. 6 89. 108.5 91. 93. 2319 m

95. $347.94 97. (a) 77; (b) �41 99. 381 ft 101.103. or 104. 105.106. 6c, or 107. or 108.109. False; 111. True 113. True


1. 20; 10; 0; �10; �20; �30 2. �18 3. �100 4. �80

5. 6. �30.033 7. 8. �10; 0; 10; 20; 30

9. 27 10. 32 11. 35 12. 13. 14. 13.455

15. �30 16. 30 17. 0 18. 19. 0

20. 0 21. �30 22. �30.75 23. 24. 120

25. �120 26. 6 27. 4; �4 28. 9; �9 29. 48; 4830. 55�C


1. �8 3. �48 5. �24 7. �72 9. 16 11. 4213. �120 15. �238 17. 1200 19. 98 21. �72

23. �12.4 25. 30 27. 21.7 29. 31.

33. �17.01 35. 37. 420 39. 41. �60

43. 150 45. 47. 1911 49. 50.4 51.

53. �960 55. 17.64 57. 59. 0 61. �720

63. �30,240 65. 1 67. 16, �16; 16, �1669. 441; �147 71. 20; 20 73. �2; 2 75. �20 lb77. �54�C 79. $12.71 81. �32 m 83.

85. 2 86. 87. True 88. False 89. False

90. False 91. (a)

�

13

DW

�5

784

10189

�2

45

27

�5

12

112

�25

�53

�83

23

2063

�7

10�

59

3 � 0 � 0 � 3y � xs � rr � s,c � 6

a � ht � 417 � yy � 7,

DW

14

�

4360

�

148

�2

15

�4130

67

�12

18

�1712

112

anspgs001-010 1/19/06 2:28 PM Page A-2

ISB

N:0-536-47742-6


93.


1. �2 2. 5 3. �3 4. 8 5. �6 6.

7. Not defined 8. 0 9. 10. 11.

12. �5 13. 14.

15.

16. 17. 18.

19. 20. 21. 22.

23. 24. �7 25. 26.

27. 28. �3.4�F per minute


1. �4 2. �0.3 3. �12 4. �9.5 5. �12 6. 2.77. �2 8. �5.7 9. �32 10. �1.8 11. 3512. 14.44 13. �2 14. �0.8 15. 1.4 16. 4


1. �8 3. �14 5. �3 7. 3 9. �8 11. 2

13. �12 15. �8 17. Not defined 19. 0 21.

23. 25. 27. 29. �7.1 31.

33. 4y 35. 37. 39.

41. 43. 45. �3x � 4� � 15�x � y13.9 � ��

11.5�

8 � ��1

13�4 � � 117�3b

2a

qp

14.3

113

�1347

715

�103

�103

,

8�7

�87

,�56

5�6

,167

�125

�2021

�5 � � 17��23 � a

�a � b� � � 17�5 � ��

18�4

7� ��

53�

23

11.6

�13

�45

32

�307

A-3

Chapter 1

47. 49. 51. 53.

55. 57. �2 59. 61. �16.2 63. Not defined

65. 44.3% 67. �5.1% 69. 71. 72. 5

73. �42 74. �48 75. 8.5 76. 77.

78. 79. �10.5, the reciprocal of the

reciprocal is the original number81. Negative 83. Positive 85. Negative


1.

2.

3. 4. 5. 6. 7. 8. 9. 1; 1

10. �10; �10 11. 12. qp13. or or 14. 19; 19 15. 150; 15016. 17.18. answers may vary19. answers may vary20. (a) 63; (b) 63 21. (a) 80; (b) 80 22. (a) 28; (b) 2823. (a) 8; (b) 8 24. (a) �4; (b) �4 25. (a) �25; (b) �2526. 5x, �8y, 3 27. �4y, �2x, 3z 28.

29. 30. 31.

32. 33.34. Associative law of multiplication35. Identity property of 136. Commutative law of addition37. Distributive law of multiplication over addition38. Identity property of 039. Commutative law of multiplication40. Associative law of addition41. 42. 43.

44. 45.

46. 47. 3x 48. 6x 49. �8x�4�3x � 8y � 4z�

18

�3x � 5y � 7�2�8a � 18b � 21�

b�x � y � z�3�x � 2y � 3�6�x � 2�

�5x � 10y � 20z5x � 10y � 20z

�2x � 635

p �35

q �35

t5x � 5

3x � 15

s � �r � 2�;�r � s� � 2,�2 � r� � s,t�4u�;�tu�4,�4t�u,

�9a�b�r � s� � 7t � yxyx � t,t � xy,

9 � x

34q

x8

�43

34

3t4t

68

1�10.5

;0.916

�0.09�18

�

14

DW

1113

964

914

53

�98

�5a � b� � 15a � b�

0

2x�x�y 2y3x

x

x � yx � yx � 2y

y

NUMBER

0

1

�8

�4.5

�54

23

OPPOSITE

0

�1

8

4.5

54

�

23

RECIPROCAL

Not defined

1

�1

4.5

�18

�45

32

Value 2x

6 6

�12 �12

9.6 9.6x � 4.8

x � �6

x � 3

x � x

Value 5x

8 10

�24 �30

19.2 24x � 4.8

x � �6

x � 2

x � 3x

anspgs001-010 1/19/06 2:28 PM Page A-3

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42-6


50. 0.59x 51. 52.

53.


1. 3. 5. 7. 9. 11.

13. 15. nm 17. or 19. or 21. 23.25. 27.29.answers may vary 31.

answers may vary33. answers may vary35. answers may vary 37.39. 41. 43.

45. 47. 49.

51. 53.

55. 57.59. 61. 4x, 3z 63. 7x, 8y, �9z65. 67. 69.71. 73.75. or 77.79. 81.83. or

85. 87. 19a 89. 9a 91.

93. 95. 97.

99. b 101. 103. 8x 105. 5n 107. �16y

109. 111. 113.

115. 117. 119.

121. 180 122. 123. True 124. False

125. True 126. True 127. Not equivalent; 129. Equivalent; commutative law

of addition 131.


1. 2. 3. 4.5. 6. 7.8. 9. 10. 11.12. 13. 14.15. 16. 17. 2 18. 18 19. 620. 17 21. 22. �1237 23. 8 24. 425. 317 26. �12


1. �11 2. 9 3. 114 4. 117,649 5. �1,419,8576. �1,124,864 7. �117,649 8. �1,419,8579. �1,124,864 10. �4 11. �2 12. 787

5x � y � 8�18.6x � 19y3x � 7

�26a � 41b � 48c�16a � 18�9x � 8y�2a � 8b � 3c3y � 32x � 73y � 2

2x � 9�18 � m � 2n � 4z4a � 3t � 10�x � y�6 � t�5x � 2y � 8�x � 2

q�1 � r � rs � rst�3 � 2 � 5 � 3 � 5 � 2

413

DW356

a �32

b � 420.8x � 0.5y

7x � y4x � 2y17a � 12b � 1

134

y

4t � 6y � 4�19a � 887x � 15y2

8x � 9z13

�2x � 5y � 1�

�6�3x � 2y � 1�6��3x � 2y � 1�,a�x � y � z�a�x � 1�

2�4x � 5y � 11��4� y � 8�4��y � 8�,8�x � 3�5�x � 2 � 3y�

7�2x � 3y�5�6 � y�2�x � 2��3.72x � 9.92y � 3.41

�4x � 12y � 8z45x � 54y � 72

�35

x �35

y � 67.3x � 14.6

23

b � 4�3x � 217x � 21

7x � 28 � 42y30x � 127 � 7t2b � 10�7b�a;b�7a�,a�7b�,

3� yx�;y�x � 3�,�3x�y,�w � v� � 5;

�v � 5� � w;�5 � w� � v;�b � 2� � a;�2 � a� � b,2 � �b � a�,

�3a�ba � �b � 3�8�xy��a � b� � 2ba � cc � ab,

9 � yxxy � 9,8 � y

4s3

�76

�32

2xx2

10x15x

3y5y

�23

�1

10x �

79

y

�4x � 5y � 73x � 3y

A-4

Answers


1. 3. 5.7. 9. 11.13. 15. 17. 19.21. 23. 25.27. 7 29. �40 31. 19 33. 35.37. 39. 41. �7 43. �745. �16 47. �334 49. 14 51. 1880 53. 1255. 8 57. �86 59. 37 61. �1 63. �1065. �67 67. �7988 69. �3000 71. 60 73. 1

75. 10 77. 79. 81. �122 83.

85. Integers 86. Additive inverses87. Commutative law 88. Identity property of 189. Associative law 90. Associative law91. Multiplicative inverses 92. Identity property of 093. 95.97. 99. (a) 52; 52; 28.130169; (b) �24; �24; �108.307025 101. �6

Concept Reinforcement, p. 76

1. True 2. True 3. False 4. True 5. False6. True 7. False

Summary and Review: Chapter 1, p. 76

1. 4 2. or 3. �45, 72 4. 385.

6. 7. 8.

9. 10. 11. �3.8 12. 13. 14.

15. 34 16. 5 17. �3 18. �4 19. �5 20. 1

21. 22. �7.9 23. 54 24. �9.18 25.

26. �210 27. �7 28. �3 29. 30. 40.4

31. �2 32. 2 33. �9 34. 8-yd gain 35. �$13036. $4.64 37. $18.95 38. 39.40. 41. 42.43. or 44.45. or 46.47. 48. 49. 50.51. 52. 6 53. 54.55. 56. True 57. False 58.59. If the sum of two numbers is 0, they are opposites, or additive inverses of each other. For every real number a, the opposite of a can be named �a, and

60. No; and 0

is not positive. 61. 62. �2.1 63. 1000

64. 4a � 2b

�58

�0� � 0,DWa � ��a� � ��a� � a � 0.

DWx � �3�15x � 25

5x � 2412y � 34�2b � 21�3a � 9�a � 8b5x � y�2x � 5y

7a � 3b�3�x � 4y � 4�3��x � 4y � 4�,5�x � 2��6�x � 1�6��x � 1�,

2�x � 7��24 � 48x4x � 15�8x � 1015x � 35

34

�27

�75

�17

83

34

��

��

0

�89

1�1�2�3�4�5�6 2 3 4 65

�2.5

0 1�1�2�3�4�5�6 2 3 4 65

0.19x19%x,

�2x � f6m � ��3n � 5m � 4b�6y � ��2x � 3a � c�

DW�2318

�1345

�4x � 649x � 183x � 3012x � 30

37a � 23b � 35c�7x � 10y9y � 25z�19x � 2y5x � 6�3a � 95x � 3

8x � 6y � 43�3x � 5y � 6�6x � 8y � 5�4a � 3b � 7c�8 � x�2x � 7

anspgs001-010 1/19/06 2:28 PM Page A-4

ISB

N:0-536-47742-6


A-5

Chapter 1

A-5

Answers

Test: Chapter 1, p. 79

1. [1.1a] 6 2. [1.1b] 3. [1.1a] 240 4. [1.2d] 5. [1.2d] 6. [1.2d] 7. [1.2d]

8. [1.2e] 7 9. [1.2e] 10. [1.2e] 2.7 11. [1.3b]

12. [1.3b] 1.4 13. [1.3b] 8 14. [1.6b]

15. [1.6b] 16. [1.4a] 7.8 17. [1.3a] �8

18. [1.3a] 19. [1.4a] 10 20. [1.4a] �2.5

21. [1.4a] 22. [1.5a] �48 23. [1.5a]

24. [1.6a] �9 25. [1.6c] 26. [1.6c] �9.728

27. [1.8d] �173 28. [1.8d] �5 29. [1.4b] 14�F30. [1.3c], [1.4b] Up 15 points 31. [1.5b] 16,080

32. [1.6d] 33. [1.7c]

34. [1.7c] 35. [1.7d] 36. [1.7d] 37. [1.4a] 1238. [1.8b] 39. [1.8b] 40. [1.8c] 41. [1.8d] �4 42. [1.8d] 44843. [1.2d] 44. [1.2e], [1.8d] 1545. [1.8c] 46. [1.7e]

CHAPTER 2


1. False 2. True 3. Neither 4. Yes 5. No6. No 7. Yes 8. 9 9. 10. 22 11. 13.212. 13. 14.


1. Yes 3. No 5. No 7. Yes 9. No 11. No13. 4 15. 17. 19. 21. 1523. 25. 2 27. 20 29. 31. 33. 19.935. 37. 39. 41. 43. 5.1 45. 12.4

47. 49. 51. 53. 55. 56. 5

57. 58. 59. 60. 61.62. 172.72 63. 64. miles 65. 342.24667. 69. 71. All real numbers 73.75. 13,


1. 15 2. 3. 4. 10 5. 10 6.7. 7800 8. 9. 28


1. 6 3. 9 5. 12 7. 9. 1 11. 13.15. 6 17. 19. 36 21. 23. 25.27. 29. 7 31. 33. 8 35. 15.9 37. �50�79

2

�32�

35�21�63

�6�7�40

�3�

45�18�

74

�13�

517�10�

2615

65t$83 � x�

124�5.2�

32

13�

512

�11DW�10211 5

6�5

�1

204124�

74

73

6 12�6�14

�18�14�20

318�2�6.5

�13

4x � 4y4a�2 � x68y � 8

9a � 12b � 72x � 77�x � 3 � 2y�

2�6 � 11x��5y � 5

18 � 3x3335

�C per minute

34

316

78

740

74

�12

�23

94

��ft2x � 9

A-5

Chapters 1–2

39. 41. 43. 44. 45.46. 47. 48. 49.50. 51. miles 52. or 53. 55. No solution 57. No solution

59. 61.


1. 5 2. 4 3. 4 4. 39 5. 6. 7.8. 800 9. 1 10. 2 11. 2 12. 13.14. or 15. 2 16. 3 17. 18.19. Yes 20. Yes 21. Yes 22. Yes 23. No24. No 25. No 26. No 27. All real numbers28. No solution


1. Left to the student 2. Left to the student


1. 5 3. 8 5. 10 7. 14 9. 11. 13.15. 17. 6 19. 4 21. 6 23. 25. 127. 6 29. 31. 7 33. 2 35. 5 37. 239. 10 41. 4 43. 0 45. 47. 49.51. 53. 55. 57. 59. 6 61. 263. No solution 65. All real numbers 67. 6 69. 871. 1 73. All real numbers 75. No solution77. 17 79. 81. 83. 2 85.87. No solution 89. All real numbers 91.93. 95. 96. 97.98. 99. 100.101. 102. 0.25 103. 105.


1. 2.8 mi 2. 280,865 socks 3. 341 mi 4.

5. 6. 7. 8.

9. 10. or 11.

12. 13. 14.

15.


1. (a) 57,000 Btu’s; (b) 3. (a) (b)

5. (a) 1423 students; (b)

7. 10.5 calories per ounce 9. 42 games 11.

13. 15. 17. x � y � bx � y � 13c �ab

x �y5

n � 15f

t � 5M1 35 mi;a �

B30

c � 4A � a � b � d

D �C

Q �a � p

tx �

y � bm

p �bqa

59

xy �5x9

,x � y � b

x � y � 7x � y � 5I �ER

r �dt

q � 3B

5245�

53291x � 242

�17x � 18�1608� y � 11x � 1�7�x � 3 � 2y��75.14�6.5DW

�5131

47�3�

53

�2827

45�4�2

25�

43�1

�20�32

3

�7�8�8

�12�2�4.3�

4310 ,

83

172

�3�4.3�32

4ba

b3a

�86555b m21

2 b � 10 m2,8r�22a � 4�10y � 42�5x � 23x � 4�32y

8x � 11�x � 57xDW�14

anspgs001-010 1/19/06 2:28 PM Page A-5

ISB

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-477

42-6


19. 21. 23. or

25. 27. 29.

31. 33. 35.

37. or 39.

41. 43. 45. 47.

49. 51. 1 52. 53. 54. 4455. 56. 57. 58.59. (a) 1901 calories;

(b)

61. or or

63. quadruples. 65. increases by units.


1. 2. 3.4. 5. 6.7. 1.92 8. 115 9. 36% 10.11. About 1.2 million 12. About 58%


1. 20% 3. 150 5. 546 7. 24% 9. 2.5 11. 5%13. 25% 15. 84 17. 24% 19. 16% 21.23. 0.8 25. 5 27. 40 29. $198 31. $158433. $528 35. Japan: 44.3%; Germany: 24.9%37. About 603 at-bats 39. $195 41. (a) 16%; (b) $2943. (a) $3.75; (b) $28.75 45. (a) $28.80; (b) $33.1247. 200 women 49. About 31.5 lb 51. $655; 196%53. $2190; $1455 55. $3935; 261% 57.59. 60. 61. 62. 2 63.64. 65. 66. 67. Division;subtraction 68. Exponential; division; subtraction69. 6 ft 7 in.


1. 2. Jenny: 8 in.; Emma: 4 in.; Sarah: 6 in.3. 313 and 314 4. 60,417 copies5. Length: 84 ft; width: 50 ft 6. First: second: third: 7. Second: 80; third: 89 8. $8400 9. $658

Translating for Success, p. 133

1. B 2. H 3. G 4. N 5. J 6. C 7. L 8. E9. F 10. D

60�90�;30�;

62 23 mi

�6 18�3.97x � 9y

a � c� 25�100�11

DW

46 23

111,416 mi2n � 94 � 10.516 � n � 80110% � b � 3043 � 20% � ba � 60% � 7013% � 80 � a

2hAA

2H

� ba �2 � Hb

H,a �

2H

;b �Ha � 2

H,

w �K � 917 � 6h � 6a

6

h �K � 917 � 6w � 6a

6;

a �917 � 6w � 6h � K

6;

�32

16�21a � 12b�13.2

�9.325�90DW

t �3kv

x �c � By

Ac2 �

Em

a �Fm

a � 2A � b12

P � lw �P � 2l

2,

h �Ab

t �A � b

ab � 3A � a � c

x �y � c

bt �

W � bm

x �ByA

58

xy �5x8

,x � a � yx � 5 � y

A-6

Answers


1. 180 in.; 60 in. 3. $4.29 5. $6.3 billion 7.9. 1204 and 1205 11. 41, 42, 43 13. 61, 63, 6515. Length: 48 ft; width: 14 ft 17. $75 19. $8521. 11 visits 23. 25.27. $350 29. $852.94 31. 12 mi 33. $3635. $25 and $50 37. 39. 40.41. 42. 43. 44. 1.6 45. 409.646. 47. 48. 0.1 49. 120 apples51. About 0.65 in. 53. $9.17, not $9.10


1. (a) No; (b) no; (c) no; (d) yes; (e) no; (f ) no2. (a) Yes; (b) yes; (c) yes; (d) no; (e) yes; (f ) yes

�41.6�9.6�10�

3215�

310

�1740�

4740

DW

109�38�,33�,68�84�,28�,

699 13 mi

3. 4.

0�2

x � �2

0 4

x 4

5. 6.

0 2

�x � x � 2�;

0�2 4

�2 � x 4

7. 8.

0�3

�x � x � �3�;

0 3

�x � x � 3�;

9. 10. � y � y � �3��x x �2

15�11. 12.

0 10

32

� y � y � 32�;

0 8

�x � x � 8�;

13. 14.

15. 16. 17.

18. 19. 20.


1. (a) Yes; (b) yes; (c) no; (d) yes; (e) yes3. (a) No; (b) no; (c) no; (d) yes; (e) no

�x x �83��x � x � �4��x � x � �2�

� y y �199 �� y y �

199 ��x x � �

14�

� y y � �135 ��x � x � �6�

5. 7.

0�3

t � �3

0 4

x � 4

9. 11.

0�3 4

�3 � x 4

0�1

m � �1

13. 15.

0�5

�x � x � �5�;

0 3

0 � x � 3

17. 19. � y � y � �5�

0�20

�18

�x � x � �18�;

21. 23. 25.27. 29. 31. �x x �

712�� y y �

14��t � t � 14�

�x � x � 4��x � x � �3��x � x � 2�

33. 35.

0 3

�x � x � 3�;

70

�x � x � 7�;

37. 39. 41.43. 45. 47.49. 51. 53.55. 57. 59. �x � x � �10��x � x � 7��x � x � �3�

�x � x � �3��x � x � 6��x � x � 8��x x �

310�� y y � �

114��x x �

173 �

� y � y � 4��x � x � �6�� y y � �25�

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A-7

Chapter 2

61. 63. 65.67. 69. 71.73. 75. 77.79. 81. 83.85. 87. 88. 4.8 89. 90.91. 92. 93. 94. 1.11 95. 14096. 41 97. 98. 99. (a) Yes; (b) yes; (c) no; (d) no; (e) no; (f ) yes; (g) yes101. No solution


1. 2. 3.4. 5. 6. 7.8. 9. 10.11.

12.


1. 3. 5.7. 9. 11. 13.15. 17. 19.21. 23. 25.27. 29. 15 or fewer copies 31. 5 min ormore 33. 2 courses 35. 4 servings or more37. Lengths greater than or equal to 92 ft; lengths less thanor equal to 92 ft 39. Lengths less than 21.5 cm41. The blue-book value is greater than or equal to $10,625.43. It has at least 16 g of fat. 45. Dates at least 6 weeksafter July 1 47. Heights greater than or equal to 4 ft49. 21 calls or more 51. 53. Even 54. Odd55. Additive 56. Multiplicative 57. Equivalent58. Addition principle 59. Multiplication principle; is reversed 60. Solution61. Temperatures between and 63. They contain at least 7.5 g of fat per serving.


1. True 2. True 3. True 4. False 5. True6. False


1. 2. 1 3. 25 4. 9.99 5. 6. 7 7.8. 9. 10. 11. 4 12. 13.14. 3 15. 4 16. 16 17. All real numbers 18. 619. 20. 28 21. 4 22. No solution 23. Yes24. No 25. Yes 26. 27.28. 29. 30.

31. 32. 33. �x x � �1

12��x x � �9

11�� y � y � �7��x � x � �11�� y � y � �4�� y � y � 2�

�x � x � 7�� y y � �12�

�3

�13�5�8�

1564�

73

�19214�22

�9 49 � C�15�C

DW

�L � L � 5 in.��Y � Y � 1935��C � C � 1063��x � x � 84�

3x � 2 � 13A � 500 Ln � 1300y � �4x � 8c � $1.50a � 1,200,000

90 mph � s � 110 mphw � 2 kgn � 7

�s � s � 94�91 � 86 � 89 � s

4� 90;

�C C � 31 19��9

5 C � 32 � 88;s � 23d � 11.4%c � 12,500

n � �2w � 110d � 1545 � t � 55p � 21,900c � 4000m � 92

37x � 1�2x � 23�9.4�

78�38

�1.11�58�74DW

�x � x � �2��x x � �5734��r � r � �3�

�t t � �53��m � m � 6�� y � y � 6�

� y � y � �3��x � x � 9��x � x � �4�� y � y � �2�� y � y � 3�� y � y � 2�

34. 35.

0

�2 � x 5

�2 50

x � 3

3

36. 37. 38.

39. 40. 41. Length: 365 mi;

width: 275 mi 42. 345, 346 43. $211744. 27 appliances 45. 46. 1547. 18.75% 48. 600 49. About 26% 50. $22051. $53,400 52. $138.95 53. 86 54.55. The end result is the same either way. If is theoriginal salary, the new salary after a 5% raise followed by an8% raise is . If the raises occur the other wayaround, the new salary is . By the commutativeand associative laws of multiplication, we see that these areequal. However, it would be better to receive the 8% raisefirst, because this increase yields a higher salary initiallythan a 5% raise. 56. The inequalities are equivalentby the multiplication principle for inequalities. If wemultiply both sides of one inequality by the otherinequality results. 57. 23, 58. 20,

59.


1. [2.1b] 8 2. [2.1b] 26 3. [2.2a] 4. [2.2a] 495. [2.3b] 6. [2.3a] 2 7. [2.3a] 8. [2.1b]

9. [2.3c] 7 10. [2.3c] 11. [2.3b] 12. [2.3c] No solution 13. [2.3c] All real numbers14. [2.7c] 15. [2.7c]

16. [2.7d] 17. [2.7d]

18. [2.7d] 19. [2.7d]

20. [2.7e] 21. [2.7e] �x � x � �1��x � x � �6��x x � �

120�� y � y � 8�

� y � y � �13��x � x � 5��x � x � �13��x � x � �4�

52

53

�7

20�8�12�6

a �y � 32 � b

�20�23�1,

DW

1.05�1.08s�1.08�1.05s�

sDW�w � w � 17 cm�

60�85�,35�,

x �y � b

ma � 2A � b

B �3Vh

d �C

0

y � 0

22. [2.7b] 23. [2.7b, e]

0 1

x � 1

0 4 9

y 9

24. [2.7b] 25. [2.5a] 18

26. [2.5a] 16.5% 27. [2.5a] 40,00028. [2.5a] About 53.4% 29. [2.6a] Width: 7 cm; length:11 cm 30. [2.5a] About $240.7 billion31. [2.6a] 2509, 2510, 2511 32. [2.6a] $88033. [2.6a] 3 m, 5 m 34. [2.8b] 35. [2.8b] 36. [2.8a]

37. [2.4b] 38. [2.4b]

39. [2.4b] or

40. [1.2e], [2.3a] 15, 41. [2.6a] 60 tickets�15

ca � 1c

d �1 � ca

�c,

x �y � b

8r �

A2h

�c � c � 143,750��b � b � $105��l � l � 174 yd�

0�2 2

�2 x 2

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CHAPTER 3


1.–8.

9. Both are negative numbers. 10. First, positive; second, negative 11. I 12. III 13. IV 14. II15. On an axis 16.

17. No18. Yes 19. answers may vary


1. 3. II 5. IV 7. III9. On an axis 11. II13. IV 15. I17. Positive 19. II21. I, IV 23. I, III

25.27. No 29. No 31. Yes33.

TRUE

TRUE

35.

TRUE

TRUE2�1 � 3

2 ? 12 ��2� � 3

y � 12 x � 3

52 � 3

5 ? 12 � 4 � 3

y � 12 x � 3

�4�4 ? 1 � 5

y � x � 5

�1�1 ? 4 � 5

y � x � 5

E : �2, 0�D: ��1, �1�;C : ��5, 0�;B: �0, �4�;A: �3, 3�;

(2, 5)

(0, 4)

(5, 0)

(0, �5)

(3, �2)

(�2, �4)

(�1, 3)

(�5, 0)

Firstaxis

Secondaxis

�1, 3�;��2, �3�,G: ��5, �4�F : �0, �3�;E : �1, 0�;D: �3, 3�;

C: �0, 4�;B: ��3, 2�;A: ��5, 1�;

(5, �3)(�3, �4)

(0, �3)

(5, 4)

(2, 0)

(4, 5)

(�2, �1)

(�2, 5)

Firstaxis

Secondaxis

A-8

Answers

37.

TRUE

TRUE

39. 41. 8 42. 43. All real numbers

44. No solution 45. $3.57 46. $48.60 47. 20%48. $18 49. $45.15 50. $55 51.53. 55. 26 linear units

57. A: B: C: D: E: F: G: H: I: J: K: 59. The figure is translated 3 units down.


1.

2.

x

y

y � qx

x

y

y � �2x

�12, 15��17, 13�;�12, 11�;�17, 8�;�12, 5�;�8, 5�;�3, 8�;�8, 11�;�3, 13�;�8, 15�;�10, 18�;

(0, 6)(1, 5)

(2, 4)(3, 3)

(4, 2)(5, 1)

(6, 0)

(�1, 7)

Firstaxis

Secondaxis

��1, �5�

74

DW

10 16 � 6

4 � 4 � 2 � 3 ? 10

4x � 2y � 10

10 0 � 10

4 � 0 � 2��5� ? 10

4x � 2y � 10

x

y

y � x � 5

(3, �2)

x

y

y � qx � 3

(�4, 1)

x

y

4x � 2y � 10

(1, �3)

x y

�3 6

�1 2

0 0

1 �2

3 �6 �3, �6�

�1, �2�

�0, 0�

��1, 2�

��3, 6�

(x, y)

x y

4 2

2 1

0 0

�2 �1

�4 �2

�1 ��1, � 12��

12

��4, �2�

��2, �1�

�0, 0�

�2, 1�

�4, 2�

(x, y)

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3. 4.

5. 6.

7. 8.

9. 10.

11. (a) $2720; $2040; $680; $0; (b) about $1700; (c) about 2.8 yr


1. Left to the student


1. 2. y �3x � 1

�10 10

�10

10

�10 10

�10

10

y 2x � 1

v � �0.68t � 3.4

t

v

2 4 6 8

2

4

1

3

$5

Time from date of purchase(in years)

Val

ue

of c

op

ier

(in

th

ou

san

ds)

x

y

5y � 3x � 20(0, 4)

x

y

5y � 3x � �10

(0, �2)

x

y

4y � 3x

(0, 0)x

y

5y � 4x � 0

(0, 0)

x

y

(0, �1)y � �Ex � 1

x

y

y � Ex � 2 (0, 2)

x

y

y � �qx � 3

x

y

y � 2x � 3

A-9

Chapter 3

3. 4.

5. 6.

7. 8.


1.

3.

x

y

y � x

(0, 0)

x

y

y � x � 1(0, 1)

�10 10

�10

10

y �3.45x � 1.68

�10 10

�10

10

y 2.085x � 5.08

y ��x � 135

�10 10

�10

10

y �x � 245

�10 10

�10

10

y 4x � 5

�10 10

�10

10

y �5x � 3

�10 10

�10

10

x y

�2 �1

�1 0

0 1

1 2

2 3

3 4

x y

�2 �2

�1 �1

0 0

1 1

2 2

3 3

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42-6


5.

7. 9.

11. 13.

15. 17.

19. 21.

x

y

(0, 5)

8x � 2y � �10

x

y

y � wx � 1

(0, 1)

x

y

(0, 4)

x � 2y � 8

x

y

y � fx � 2

(0, �2)

x

y

x � y � �5

(0, �5)

x

y

y � qx � 1

(0, 1)

x

y

y � 3x � 2

(0, �2)

x

y

y � x � 3

(0, �3)

x

y

(0, 0)

y � qx

A-10

Answers

23.

25. (a) $300, $100, $0; (b) $50;(c) 3 yr

27. (a) 22 gal, 25.2 gal, 30 gal, 33.2 gal;(b) 27 gal; (c) in 13 yr, or in 2008

29. 31. 12 32. 4.89 33. 0 34. 35. 3.4

36. 37. 38. 39. $16.81 40. $18.40


1. (a) (b)2. 3.

4. 5.

x

y

y ��x23

x

y

y � 2x

x

y

3y � 4x � 12

(0, 4)

(�3, 0)

x

y

2x � 3y � 6

(0, 2)

(3, 0)

�4, 0��0, 3�;

78

232

45

DW

N � 0.8d � 21.2

d

N

5 10 15 20

30

25

20

15

35

V � �50t � 300

t

V

2 4 6 8 10

100

200

300

400

x

y

(0, �q)

8y � 2x � �4

x y

�2 �1

0 0

4 2

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6. 7.

8. 9.


x

y

x � �3

x

y

x � 0

x

y

y � �2

x

y

x � 5

A-11

Chapter 3


1. (a) (b) 3. (a) (b)5. (a) (b) 7. (a) (b)9. (a) (b) 11. (a) (b)13. 15.

17. 19.

21. 23.

25. 27.

29. 31.

x

y

3x � 2 � y

(s, 0)

(0, �2)x

y

x � 3 � y

(3, 0)

(0, �3)

x

y

2x � 3y � 8 (0, h)

(4, 0)(5, 0)

(0, 4)

x

y

4x � 5y � 20

(0, �2)(3, 0) x

y

2x � 3y � 6

x

y

3x � 9 � 3y

(0, �3)

(3, 0)

x

y

2y � 2 � 6x

(�a, 0) (0, 1)(2, 0)

(0, 6)

x

y

3x � y � 6

(�4, 0)

(0, 2)

x

y

�x � 2y � 4

(0, 2)

(6, 0) x

y

x � 3y � 6

�12 , 0��0, �

13�;��

52 , 0��0, 10

3 �;�4, 0��0, �14�;�5, 0��0, 3�;

�3, 0��0, �4�;�2, 0��0, 5�;

1. y-intercept: x-intercept:


Xscl � 5 Yscl � 5

�25 10

�10

50

y � 2.15x � 43

��20, 0�;�0, 43�;

�5 5

�25Xscl � 1 Yscl � 5

10

y � �7.5x � 15

��2, 0�;�0, �15�;



y � 0.2x � 4

�5 30


5

�20, 0�;�0, �4�;

y � (6x � 150)�5

�10 35


10

�25, 0�;�0, �30�;


6. y-intercept:

x-intercept:

y � (5x � 2)�4

�1 1

�1

1

Xscl � 0.25 Yscl � 0.25

�25 , 0�;

�0, �12�;


�5 20

�25

5

y � 1.5x � 15

�10, 0�;�0, �15�;

Visualizing for Success, p. 191

1. E 2. C 3. G 4. A 5. I 6. D 7. F 8. J9. B 10. H

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42-6


33. 35.

37. 39.

41. 43.

45. 47.

49. 51.

53. 55.

x

y

48 � 3y � 0

�8 �4 4 8

4

8

12

x

y

4x � 3 � 0

x

y

3y � �5

x

y

x � w

x

y

y � 0

x

y

x � 2

x

y

y � 2

x

y

x � �2

x

y

y � 3x � 0

(0, 0)x

y

y � �3 � 3x

(�1, 0)

(0, �3)

x

y

3x � 4y � 5

(f, 0)

(0, @)

x

y

6x � 2y � 12

(2, 0)

(0, �6)

A-12

Answers

57. 59. 61. 63. 16%64. $32.50 65. 66.67. 68. 69.71.


1. 2.

3. 4. 4.5 cents per minute5. �1700 deaths by firearms per year 6. 4 7. �178. �1 9. 10. �1 11. 12. Not defined 13. 0


1. This line will pass through the origin and slant up fromleft to right. This line will be steeper than 2. This line will pass through the origin and slant up fromleft to right. This line will be less steep than


1. This line will pass through the origin and slant downfrom left to right. This line will be steeper than 2. This line will pass through the origin and slant downfrom left to right. This line will be less steep than


1. 3. 5. 7. 0

9. 11. 3;

x

y

(�4, 0)

(�5, �3)

x

y

(�2, 4)

(3, 0)

�45 ;

34

23�

37

y � �5

32 x.

y � �10x.

y � 532 x.

y � 10x.

54

23

63 711 %, or 63.63%

x

y

(�3, 2)

(0, �3)

�53

x

y

(�2, 3)(3, 5)

25

k � 12y � �4�x � x � 2��x � x � 1�

�x � x � �7��x � x � �40�DWx � 4y � �1

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13. 15.

17. 19. Not defined 21. 23. 0 25.27. 29. About 29.4% 31. 25 miles per gallon33. �$500 per year 35. About 7600 people per year37. �10 39. 3.78 41. 3 43. 45.47. Not defined 49. �2.74 51. 3 53. 55. 0

57. 59. Equivalent equations

60. Addition principle 61. Multiplication principle62. Horizontal 63. Vertical 64. Slope65. x-intercept 66. y-intercept67. 69.71.

73.


1. True 2. False 3. True 4. True 5. False6. True


1. 2. 3.4.–6. 7. IV 8. III

9. I 10. No11. Yes

x

y

(�4, �2)(0, �3)

(2, 5)

�3, 0��2, 5��5, �1�

y � x 3 � 5

�10

�10

10

10

�1005�975.3�946.2�917.7�889.7�862.4�835.6

X

X � �10

Y1

�10�9.9�9.8�9.7�9.6�9.5�9.4

y � 0.35x � 7

�10

�10

10

10

�10.5�10.47�10.43�10.4�10.36�10.33�10.29

X

X � �10

Y1

�10�9.9�9.8�9.7�9.6�9.5�9.4

y � x � 2y � �x � 5

DW

54

�32�

15

28129

1241�

513

23

x

y

(5, 3)

(�3, �4)

78 ;

x

y

(2, �3)

(�4, 2)

�56 ;

A-13

Chapter 3

12.

TRUE

TRUE

13. 14.

15. 16.

17. 18.

19. 20.

(0, �5)

(2, 0)

5x � 2y � 10

x

y

(0, �3)

(6, 0)

x � 2y � 6

x

y

5x � 4 � 0

x

y

y � 3

x

y

(0, 3)

y � 3 � 4x

x

y

(0, 4)

y � �x � 4

x

y

(0, 0)

y � �!x

x

y

y � 2x � 5

x

y

(0, �5)

3 4 � 1

2 2 � 1 ? 3

2x � y � 3

3 0 � 3

2 0 � ��3� ? 3

2x � y � 3

x

y

(3, 3)

2x � y � 3

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21. (a)(b)

(c) 6 residents

22. (a) 2.4 driveways per hour; (b) 25 minutes per driveway23. 4 manicures per hour 24. 25.

26. 27.

28. 7% 29. 30. 31. Not defined 32. 0

33. The y-intercept is the point at which the graphcrosses the y-axis. Since a point on the y-axis is neither leftnor right of the origin, the first or x-coordinate of the pointis 0. 34. The x-intercept is the point at which thegraph crosses the x-axis. Since a point on the x-axis isneither up nor down from the origin, the second coordinateof the point is 0. 35. 36. 45 square units; 28 linear units 37. (a) 3.709 feet per minute; (b) about0.2696 minute per foot


1. [3.1a] II 2. [3.1a] III 3. [3.1b] 4. [3.1b] 5. [3.1c]

TRUE

TRUE 5 3 � 2

3 � 2��1� ? 5

y � 2x � 5

5 �3 � 8

�3 � 2��4� ? 5

y � 2x � 5

�0, �4��3, 4�

m � �1

DW

DW

12�

58

x

y

(4, �4)

(�5, 5)

�1;

x

y

(5, 4)

(�5, �2)

35 ;

�13

13

17 12 ft3;

n

S

10

20

30

40

Number of people in household

Ref

rige

rato

r si

ze(i

n c

ub

ic fe

et)

10 20

S ��n � 1332

14 12 ft3, 16 ft3, 20 1

2 ft3, 28 ft3;

A-14

Answers

6. [3.2a] 7. [3.2a]

8. [3.3b] 9. [3.3b]

10. [3.3a] 11. [3.3a]

12. [3.2b] (a) $5000; $11,600; $14,000; $16,400;(b) $17,000;

(c) 2015

13. [3.4b] (a) 14.5 floors per minute; (b) per floor 14. [3.4b] 87.5 miles per hour 15. [3.4a] �216. [3.4a]

17. [3.4b] 18. [3.4c] (a) (b) not defined19. [3.1a] 25 square units; 20 linear units 20. [3.3b] y � 3

25 ;�

120

x

y

(5, 4)

(�3, 1)

38 ;

4 429 seconds

35

T � �n � 5

n

T

2 4 6 8 10 12 14 16 18 20

5

10

15

20

$30

25

Number of years since 1992

Co

st(i

n t

ho

usa

nd

s)

x

y

(w, 0)

(0, �3)2x � y � 3

(�4, 0)

(0, 2)

x

y

2x � 4y � �8

y � 5

x

y

x

y

2x � 8 � 0

(0, 0)

x

y

y � �wx

x

y

y � 2x � 1(0, �1)

x

y

y � 2x � 5

(�2, 1)

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Cumulative Review: Chapters 1–3, p. 217

1. [1.1a] 2. [1.7c] 3. [1.7d] 4. [1.7c] 5. [1.2c] 0.45 6. [1.2e] 4 7. [1.3b] 3.088. [1.6b] 9. [1.7e] 10. [1.3b] �24.6

11. [1.4a] 12. [1.3a] 2.6 13. [1.5a] 7.28

14. [1.6c] 15. [1.8d] �2 16. [1.8d] 2717. [1.8b] 18. [1.8c] 19. [2.1b] �1.2 20. [2.2a] �21 21. [2.3a] 922. [2.2a] 23. [2.3b] 2 24. [2.1b]

25. [2.3c] 26. [2.3b] �17 27. [2.3b] 2

28. [2.7e] 29. [2.7e]

30. [2.3c] All real numbers 31. [2.3c] No solution

32. [2.3c] All real numbers 33. [2.4b]

34. [3.1a] IV 35. [2.7b]

36. [3.3a] 37. [3.3b]

38. [3.2a] 39. [3.2a]

40. [3.2a] 41. [3.2a]

42. [3.3a] y-intercept: x-intercept:

43. [3.3a] y-intercept: x-intercept: 44. [2.5a] 160 million 45. [2.6a] 15.6 million46. [2.5a] $120 47. [2.6a] First: 50 m; second: 53 m;third: 40 m 48. [2.8b] �x � x � 8�

��54 , 0��0, 5�;

�10.5, 0��0, �3�;

x

y

y ��x32

x

y

y ��x23

x

y

y � sx � 2

x

y

y � �2x � 1

x

y

y � �2

x

y

2x � 5y � 10

0�1 2

�1 � x 2

h �2A

b � c

�x � x � �118 ��x � x � 16�

�1721

138

425

5x � 11�2y � 7�

512

13

�x � y�78

3y, �2x, 43�5x � 3y � 1�12x � 15y � 215

2

A-15

Chapters 3–4

49. [3.2b] (a) $375, $450, $525, $825; (b) $1050; (c) 196 months

50. [3.4b] per mile 51. [3.4a] 52. [3.3a] (b)53. [1.8d] (e) 54. [1.8b] (d) 55. [3.4a] (d)56. [2.3a], [1.2e] �4, 4 57. [2.3b] 3

58. [2.4b]

CHAPTER 4


1. 2. 3. 4.5. 6. 7. 68. 1 9. 8.4 10. 1 11. �1.4 12. 0 13. 12514. 160 15. 16. 119 17. 3; 18. (a) 144; (b) 36; (c) no 19. 20. 21.22. 23. 24. 25. 26. 27.

28. 29. 30.

31. 32. 33. 34. 35.

36. 37. b 38.


1. 3.5. 7. 9.11. 13. 1 15. b 17. 119. �7.03 21. 1 23. ab 25. a 27. 27 29. 1931. �81 33. 256 35. 93 37. 136 39. 10; 4

41. 43. 45.

47. 49. 51. 53. 55.

57. 59. 61. 63. 65. 67.

69. 71. 73. 75. 77.

79. 81. 83. 85. 1 87. 89.

91. 93. 95. 97. 99. 1 101.

103. 105. 107. 109. 1

111. ; ; ; ; ;

; ; 113. DW��15��2 � 25�� 1

5�2 � �1

25��5�2 � 25

�52 � �25�15��2 � 25�1

5�2 � 1255�2 � 1

2552 � 25

x 31z4x9

x21�8x�4

1m6

1166y4

86731a10

1x13x17

1x

33�7y�17�3y�12938

x781427a�5x�34�3

z ny482 � 641

a3173 �

1343

1103 �

11000

132 �

19

3629.84 ft2

�6 y y y y8 k k k�7p� �7p��2

3� �23� �2

3� �23�

��1.1� ��1.1� ��1.1� ��1.1� ��1.1�3 3 3 3

t 6175

1x752x24

p31

��2�3 � �18

124 �

116

152 �

125

143 �

164

a4b2p9y443a9b8x5p24x10310

�33215.36 cm2

�1 y y y��x� ��x� ��x� ��x�3 t t3t 3tx x x x x5 5 5 5

Q �2 � pm

p, or

2p

� m

�14

112 gal

P � �n � 3

n

P

2 4 6 8 10

1

2

3

4

8

$10

5

6

7

9

Months of service

Co

st(i

n h

un

dre

ds)

34

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115. 8 in., 4 in. 116. 228, 229 117.118. 51°, 27°, 102° 119. 120.121. 122. 123. No125. No 127. 129. 131. 1 133.135. 137.


1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

12. 13. 14. 15.

16. 17. 689,300,000,000 18. 0.000056719. 20. 21.22. 23.24. The mass of Saturn is times the mass of Earth.


1. 2. 3.4. 5. 6. 7.8.


1. 3. 5. 7. 9. 11.

13. 15. 17. 19. 21.

23. 25. 27. 29. 31.

33. 35. 37. 39. 41.

43. 45. 47. 49. 51.

53. 55. 57.59. 61. 63. 65.67. 87,400,000 69. 0.00000005704 71. 10,000,00073. 0.00001 75. 77.79. 81. 83.85. 87. Approximately 89. The mass of Jupiter is times the mass of Earth.91. 93. The mass of the sun is times the mass of Earth. 95. days 97.99. 100. 101.102. 103. 104. 2 105. 106.107.

y � x � 5

x

y

�112�

127

74�7�x � 2�

3�s � t � 8�2�2x � y � 8�9�x � 4�DW4.375 � 102

3.33 � 1051 � 10223.18 � 102

1.325 � 1014 ft33.0 � 10�215.0 � 10�42.5 � 10138.1477 � 10�13

3.38 � 1046 � 109

10�72.96 � 10810111.8 � 10�83.04 � 10�69.07 � 10172.8 � 1010

c 2d6

a4b2x6y3

z349x68y6

a8

b12

y6

4a12b816x6

y49x6

y16z6b21

a15c6

25t 6

r 8a10

b35x24y81x12y15

9x8

16x61m3n6

1a3b3a3b3x8

1t 12t 181

a18x1215626

3 � 10138 � 10�264 � 1053 � 10�63.6 � 1012

3.2 � 1059.044 � 1051.3545 � 10�4

9.5 � 101.884672 � 1011 L5.5 � 102

2.0 � 1037.462 � 10�135.6 � 10�155.23 � 108

5.17 � 10�49x8

8t 15

w 12x12

25

�

y15

27x6�

18x12�

1y24

27z24

y6x15x74

25x10

y12z616x20

y121

x32y151x12320

� 1

10,000��a4ty5x

2�128 � a � 2b�4�x � 3 � 6y�

1110

2314

25,543.75 ft2

A-16

Answers

108.

109. 111. 113. 115. 7117. , or 2.5 119. False 121. False 123. True


1. ; ; ; answers may vary2. 3. 4. 5. 21 6. 6; 7. 132 games 8. 360 ft 9. (a) 7.55 parts per million;(b) When , ; so the value found in part (a)appears to be correct. 10. 20 parts per million11. 12.13. , , 14. , , , 15. and 16. and ; and 17. and ; and

; and 11 18. 2, , , 10, 19.20. 21. 22. 23.24. 25. 26.27. 28.29.30.31. 32.33. 34. 4, 2, 1, 0; 4 35. 0, 3, 6, 5; 6

36. x 37. , , x, 38. , x 39.40.41.42. Monomial 43. None of these 44. Binomial45. Trinomial


1. 3; 2.25; 2. 44; 0; 9.28 3. 13; ; 74. ; ;


1. ; 7 3. 19; 14 5. ; 7. 5 9. 9; 111. 56; 13. 1112 ft 15. (a) 1138.34 billion kilowatt-hours, 1499.46 billion kilowatt-hours, 1950.86 billionkilowatt-hours, 2402.26 billion kilowatt-hours, 3305.06 billion kilowatt-hours; (b) left to the student17. $18,750; $24,000 19. , 4, 5, 2.75, 121. 1,820,000; 3,660,000 23. 9 words 25. 6 27. 1529. 2, , 31. 3 33. and 35. and ; and 37. and ; and ; 8 and 39. , 6 41. 5, 343. , 6, , 8, 45. 47.49. 51. 53. 55.57. 59. 61.63. 65.67. 69. 71. 12x4 � 2x �

14�5x2 � 9x13x3 � 9x � 8

x6 � x415y9 � 7y8 � 5y3 � y2 � y

x5 � 6x3 � 2x2 � x � 11516x3 �

76x2x4

34x5 � 2x � 424b5x3 � x11x3 � 4

�8x�3x�2�2.7�5

34 ,�3�9�2x

�7x14x53x5�7x5x�3x42x4�3x26x2�x,1

3 x3,�2x4,x2�3x

�4

�2

133 ;�7�12�18

�40.6�7�1�3.32�27

a4 � 10a4 � 0a3 � 0a2 � 0a � 10;2x3 � 4x2 � 22x3 � 4x2 � 0x � 2;

x3x2x0x2x310x4 � 8x �

12

�2x2 � 3x � 214t 7 � 10t 5 � 7t 2 � 147x5 � 5x4 � 2x3 � 4x2 � 36x7 � 3x5 � 2x4 � 4x3 � 5x2 � x

x5 �9

20 x4 �495�

14x3 � 4x2 � 7

4x3 � 46x25 � 3x55x3�4x3�

14x5 � 2x22x3 � 7

8x2�4�8.5�7�10�8x3x7x25x210t 3�9t 3�7t 44t 4

�x34x3�2�3y7y2�4y5126x3x2

�2y3 � 3y7 � ��7y� � ��9��9x3 � ��4x5�

C � 7.5t � 3

�4�18�104�19�7x3 � 1.115y34x2 � 3x �

54

10.4

311152.478125 � 10�1

x

y

2x � y � 8

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N:0-536-47742-6


73. 1, 0; 1 75. 2, 1, 0; 2 77. 3, 2, 1, 0; 3 79. 2, 1, 6, 4; 681.

83. , x 85. 87. None missing89. ; 91. ; 93. None missing 95. Trinomial 97. None of these99. Binomial 101. Monomial 103.105. 27 apples 106. 107. 108.

109. 110. 111. 112. 45%; 37.5%;

17.5% 113. 115. 117. 10119. , 4, 5, 2.75, 1 121. 1,820,000; 3,660,000


1. 2.3. 4. 5.6. 7.8. 9.10.11. 12.13. 14.15. 16.17. 18. Sum of perimeters:

; sum of areas: 19.


1. Yes 2. Yes 3. No 4. Yes 5. No 6. Yes


1. 3. 5. 7.9. 11.13.15.17.19. 21.23. 25. 27.29. 31.33. 35. 37.39.41. 43.45. 47. 3x � 60.06x3 � 0.05x2 � 0.01x � 1

34x3 �

12x4.6x3 � 9.2x2 � 3.8x � 23

6x4 � 3x3 � 4x2 � 3x � 4�18�x2 � 7x � 57x � 1

4x4 � 6x2 �34x � 8�4x2 � 3x � 2

�3x � 7�12x4 � 3x3 � 3x2 �32 x � 2

5x1.05x4 � 0.36x3 � 14.22x2 � x � 0.979x8 � 8x7 � 6x4 � 8x2 � 40.01x5 � x4 � 0.2x3 � 0.2x � 0.06�

12 x4 �

23 x3 � x2

6 � 12x2�2.2x3 � 0.2x2 � 3.8x � 235x2 � 3x � 302x2x2 �

112 x � 1�x � 5

�x2 � 64� ft272x213x

�x5 � 2x3 � 3x2 � 2x � 22x3 � 5x2 � 2x � 5x3 � x2 �

43x � 0.9

�8x4 � 5x3 � 8x2 � 1x2 � 6x � 22x3 � 2x � 8�14x10 �

12x5 � 5x3 � x2 � 3x

�5x4 � 3x2 � 7x � 5�4x3 � 6x � 3�x3 � x2 � 3x � 3�8x4 � 4x3 � 12x2 � 5x � 88x3 � 2x2 � 8x �

52

2x2 � 3x � 12x3 �10324x4 � 5x3 � x2 � 1

�4x5 � 7x4 � x3 � 2x2 � 4x2 � 7x � 3

�43x63�x � 5y � 21�

b �C � r

a152�2.6

58�

1724�19

DW

x4 � xx4 � 0x3 � 0x2 � x � 0x0x3 � 27x3 � 0x2 � 0x � 27

x3, x2, x0x2

A-17

Chapter 4

49. 51.53. 55. 57. ;

, or 59. or ; , or

61. 63. 65.67. 6 68. 69. 70. 5 71. 5 72. 173. 74. 75. 76.77. 79. 81.83. 85.


1. 2. 3. 4. 5. 6.7. 8. 0 9. 10.11.12. (a)

;(b) or or

13. 14.15. 16.17.18.19.20.21.


1. Correct 2. Correct 3. Not correct 4. Not correct


1. 3. 5. 7. 9. 11. 013. 15. 17.19. 21. 23.25. 27. 29.31. 33. 35.37. 39.41. 43.45. 47.

49. 51.

53. 55.57. 59.61.63. 65.67. 69. x4 � 1x9 � x5 � 2x3 � x

6t 4 � t 3 � 16t 2 � 7t � 4�1 � 2x � x2 � x4�10x5 � 9x4 � 7x3 � 2x2 � x

x6 � 2x5 � x33y4 � 6y3 � 7y2 � 18y � 64x3 � 14x2 � 8x � 1x 3 � 1

x

x2 5x

3x 15

5

3

x

x � 5

x � 3

x

x2 2x

x 2

2

1

x

x � 2

x � 1

x

x2 5x

5

x

x � 5

x2 � 7x � 6�x � 1� �x � 6�,x2 � 8x � 12�x � 2� �x � 6�,x2 � 2.4x � 10.81

x2 �2110 x � 14x2 � 20x � 25

25 � 15x � 2x2x2 � 9x2 � 7x � 12x2 � 3x � 10x2 � 9x � 1818y6 � 24y5

�6x4 � 6x36x3 � 18x2 � 3xx5 � x2�5x2 � 5x�2x2 � 10x�24x11

115 x40.03x1132x8x340x2

6x4 � x3 � 18x2 � x � 1020x4 � 16x3 � 32x2 � 32x � 163x3 � 13x2 � 6x � 206y5 � 20y3 � 15y2 � 14y � 35x4 � 3x3 � x2 � 15x � 20

6x2 � 19x � 155x2 � 17x � 12x2 � x � 20x2 � 13x � 40

y2 � 9y � 14y2 � 2y � 7y � 14,� y � 2� � y � 7�,

� y2 � 9y � 14 � y2 � 7y � 2y � 14 � y y � y 7 � 2 y � 2 7

� y � 2� � y � 7� � y � y � 7� � 2� y � 7��5x6 � 25x5 � 30x4 � 40x3

�15t 3 � 6t 28x2 � 16x7y5�8y1112x7�x5x2�x2�15x

�3y4 � y3 � 5y � 212y2 � 23y � 21y2 � 4y � 42x2 � 20x20w � 42

�x�x � 0��x�x � �10�372

392

�7

22�19

DW18z � 64�r 2 � 25�x2 � 6x � 9x2 � 3x � 9 � 3x�x � 3�2�x � 3� �x � 3�,

r 2 � 20r � 999r � 99 � 11r � r 2�r � 11� �r � 9�5x2 � 4x23

2 a � 12x4 � x3 � x2 � x11x4 � 12x3 � 9x2 � 8x � 9

TERM

�2

8x

�3x2

6x3

�7x4

COEFFICIENT

�7

6

�3

8

�2

DEGREEOF THETERM

4

3

2

1

0

DEGREEOF THE

POLYNOMIAL

4

anspgs011-020 1/19/06 2:33 PM Page A-17

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-536

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42-6


71.73. 75. 77.78. 6.4 79. 96 80. 32 81.82. 83.84. 85.

86. 87. 89.91. 5 93.

95. 0 97. 0


1. 2. 3.4. 5.6. 7. 8.9. 10.11. 12.13. 14. 15. 16.17. 18. 19.20. 21. 22.23. 24.25. 26.27. 28. represents the area of the large square. This includes all four sections. represents only two of the sections. 29.30. 31.32. 33.34. 35.36.


1. E, F 2. B, O 3. S, K 4. R, G 5. D, M 6. J, P7. C, L 8. N, Q 9. A, H 10. I, T


1. 3.5. 7. 9.11. 13. 15.17. 19.21. 23.25. 27.29. 31.33. 35.37. 39.41. 43. 45.47. 49. 51.53. 55. 57.59. 61. 63.65. 67. 69. 9 � 6x � x2a2 � a �

149x4 � 6x2 � 1

x2 � 4x � 42564 x2 � 18.494y16 � 9

x24 � 9x8 � 9x2x12 � x49x8 � 164x4 � 925m2 � 4

4x2 � 1x2 � 164y6 � 4y5 � y4 � y34x3 � 12x2 � 3x � 98x6 � 65x3 � 8

6x7 � 18x5 � 4x2 � 1213.16x2 � 18.99x � 13.953x6 � 2x4 � 6x2 � 4x5 � 3x3 � x2 � 3

964 y2 �

58 y �

25361 � x � 6x2

a2 � 14a � 49�2x2 � 11x � 62x3 � 2x2 � 6x � 6x2 � 0.01

p2 �1

164x2 � 6x � 29t 2 � 15x2 � 4x � 129x2 � 12x � 4y2 � y � 6

x4 � x3 � 2x � 2x3 � x2 � 3x � 3

x3 � 3x2 � 6x � 84x2 � 2x �

1425x2 � 5x �

14

4a2 � 6a � 4081x4 � 18x2 � 1�8x5 � 20x4 � 40x2t 2 � 16

x2 � 11x � 30A2 � B2

�A � B�29x4 � 30x2 � 2560.84 � 18.72y � 1.44y216x4 � 24x3 � 9x2

4x2 � 20x � 25a2 � 8a � 16x2 � 4x � 4x2 � 10x � 25x2 � 16x � 64

x2 �4

254x6 � 136 � 16y2x2 � 49x2 � 44x2 � 9x2 � 25

30x5 � 27x4 � 6x38 � 2x2 � 15x4x5 � 0.5x3 � 0.5x2 � 0.25x2 �

1625

�2x7 � x5 � x3t 2 � 8t � 15y6 � 4912x5 � 10x3 � 6x2 � 52x3 � 4x2 � 3x � 6

2x2 � 9x � 4x2 � 2x � 15x2 � 7x � 12

�x3 � 2x2 � 210� m3S � ��4x2 � 144� in2V � �4x3 � 48x2 � 144x� in3;75y2 � 45y23

19

x

y

y � qx � 3

100�x � y � 10a��3�3x � 15y � 5�4�4x � 6y � 9�

3�5x � 6y � 4��

34

DW2x4 � 5x3 � 5x2 �1910 x �

15

x4 � 8x3 � 12x2 � 9x � 4

A-18

Answers

71. 73.75. 77.79. 81.83. 85. 87.89. 91.93. 95. 97. 25; 4999. 56; 16 101. 103.105. 107. Lamps: 500 watts; air conditioner: 2000 watts; television: 50 watts 108. 109.

110. 111. , or

112. , or

113. 115.117. 119. 121. First row: 90,

, ; second row: 7, , , , 12, , , ; third row: 9, , , 10, , , , , 21; fourth row:

, 123. Yes 125. No


1. 2. 3. 1889 calories 4. , 3, , 1, 25. 3, 7, 1, 1, 0; 7 6. 7.8. 9.10.11.12.13.14.15.16. 17.18. 19.20. 21.22.


1. 3. 5. 240 7. 9. 3.715 liters11. 92.4 m 13. 15.17. Coefficients: 1, , 3, ; degrees: 4, 2, 2, 0; 419. Coefficients: 17, , ; degrees: 5, 5, 0; 521. 23. 25.27. 29. 31.33. 35.37. 39.41. 43.45. 47.49. 51.53.55. 57.59. 61.63. 65.67. 69. 71.73. 75.77.79. 81.83. IV 84. III 85. I 86. II

DW3x4 � 7x2y � 3x2 � 20y2 � 22y � 6a2 � b2 � 2bc � c 2

x2 � y2 � 2yz � z2x2 � 2xy � y2 � 9a2b2 � c 2d 4c4 � d24a2 � b2

3a3 � 12a2b � 12ab24a6 � 2a3b3 �14 b6

p8 � 2m2n2p4 � m4n4r 6t 4 � 8r 3t 2 � 16x2 � 2xh � h2x9y9 � x6y6 � x5y5 � x2y2

m3 � m2n � mn2 � n312 � c2d2 � c4d412x2y2 � 2xy � 2

y6x � y4x � y4 � 2y2 � 1a6 � b2c 2a4b2 � 7a2b � 106z 2 � 7zu � 3u2

�2a � 10b � 5c � 8d2ab � 2ab2 � a2b�b2a3 � 3b3a2 � 5ba � 3

3r � 7x2 � 4xy � 3y28u2v � 5uv 220au � 10av3x2y � 2xy2 � x2�a � 2b

�7�3�5�2

63.78125 in244.46 in2�145�15�1

4a2 � 25b2 � 10bc � c 29y2 � 24y � 16 � 9x216y2 � 9x2y4

4x2y4 � 9x29x4 � 12x3y2 � 4x2y416x2 � 40xy � 25y22x2 � 11xy � 15y2

3x3y � 6x2y3 � 2x3 � 4x2y2p5q � 4p3q 3 � 3pq 3 � 6q 4x5y5 � 2x4y2 � 3x3y3 � 6x2�9p4q � 9p3q 2 � 4p2q 3 � 9q 4 � 5�8s4t � 6s3t 2 � 2s2t 3 � s2t 2�5p2q 4 � 2p2q 2 � 3p2q � 6pq 2 � 3q � 5

14x3y � 7x2y � 3xy � 2y�4x3 � 2x2 � 4y � 25pq � 82x2y � 3xy

�2�3�176�7940

�6�19�10�8�8�8�2�2

�11�21�6�14�36�18�63�432�781t16 � 72t 8 � 16

a4 � 50a2 � 62530x3 � 35x2 � 15x

a �53

d �43

a �5d � 4

3

y �32

x � 6y �3x � 12

2274

�417

2827

DWt 2 � 10t � 24a2 � 2a � 1

t 3 � 164 � 96x4 � 36x812x3 � 8x2 � 15x � 1036x8 � 48x4 � 16

15t 5 � 3t 4 � 3t 39p2 � 14x4 � 2x2 �14

4x3 � 24x2 � 12x9 � 12x3 � 4x6x2 �

54 x �

256425 � 60t 2 � 36t 4

4 � 12x4 � 9x8x4 � 2x2 � 1

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87. 88.

89. 90.

91. 93.95. 97. 16 gal99. $15,638.03


1. 2. 3. 4. 5.6. 7. 8.9. 10. 11.

12. , , or 13.

14. with or


1. 3. 5. 7.9. 11.13. 15.17. 19.21. 23. 25.

27. 29. 31.

33. 35.

37. 39. 41.

43. 45. Product 46. Monomial47. Multiplication; equivalent 48.49. Trinomial 50. Quotient 51. Absolute value

52. Slope 53. 55.

57. 59.61. 63. 1


1. False 2. True 3. False 4. True 5. False6. True

�5a5 � a4b � a3b2 � a2b3 � ab4 � b52x2 � x � 3

a � 3 �5

5a2 � 7a � 2x2 � 5

x � a

DW

t 2 � 13x2 � x � 2 �10

5x � 1x3 � 6

2x2 � 7x � 4x4 � x3 � x2 � x � 1

x � 3x � 2 ��2

x � 6x � 5 �

�50x � 5

x � 23rs � r � 2s6x2 � 13x � 49x2 �

52 x � 16x2 � 10x �

32

�4x4 � 4x2 � 15t 2 � 8t � 21 � 2u � u43x4 �

12 x3 �

18 x2 � 2

4a3b18x35x3x4

2x3 � x2 � 3x � 1 �11

4x � 2

R � 11;2x3 � x2 � 3x � 1

x2 � x � 1x � 4 ��2

x � 3R �2x � 4

x � 4x � 22x2y4 � 3xy2 � 5y4x2 �

32 x �

122x2 � x �

23x2 � 3x � 2

7x4 � 8x214 x4�28p3q�7x114x2

2�nh � 2�mh � 2�n2 � 2�m22xy � �x24xy � 4y2

x

y

x � 4

x

y

8y � 16 � 0

x

y

y � �4

x

y

2x � �10

A-19

Chapter 4


1. 2. 3. 4. 5. 6. 7. 1

8. 9. 10. 11. 12.

13. 14. 8,300,000 15.16. 17. gal 18. 1019. , , , 20. , 21. 3, 2, 1, 0; 322. Binomial 23. None of these 24. Monomial25. 26.27.28. 29.30. 31. Perimeter: ; area:

32. , 33. 34.35. 36.37. 38.39. 40. 41. 4942. Coefficients: 1, , 9, ; degrees: 6, 2, 2, 0; 643.44.45. 46.47. 48. 49.

50. 51. 0, 3.75, , 0

52. is not in scientific notation because 578.6 is larger than 10. 53. A monomial is an expression of the type , where n is a whole number anda is a real number. A binomial is a sum of two monomialsand has two terms. A trinomial is a sum of three monomialsand has three terms. A general polynomial is a monomial ora sum of monomials and has one or more terms.54. 55. 56. 57.58. 59. 16 ft by 8 ft


1. [4.1d, f ] 2. [4.1d] 3. [4.1d]

4. [4.1e] 5. [4.1e, f ] 6. [4.1b, e] 1 7. [4.2a]

8. [4.2a, b] 9. [4.2a, b] 10. [4.2b]

11. [4.1d], [4.2a, b] 12. [4.1d], [4.2a, b] 13. [4.1d], [4.2a, b] 14. [4.1d], [4.2a, b]

15. [4.1f] 16. [4.1f] 17. [4.2c]

18. [4.2c] 0.00000005 19. [4.2d] 20. [4.2d] 21. [4.2e] files22. [4.3a] 23. [4.3d] , , 7 24. [4.3g] 3, 0, 1, 6; 625. [4.3i] Binomial 26. [4.3e] 27. [4.3e] 28. [4.3f]

29. [4.4a] 30. [4.4a] 31. [4.4c] 32. [4.4c] 33. [4.5b] 34. [4.6c] 35. [4.6b] 36. [4.6a] 3b2 � 4b � 159x2 � 100

x2 �23 x �

19�12x4 � 9x3 � 15x2

�x5 � 0.7x3 � 0.8x2 � 21�4x4 � x3 � 8x � 35x4 � 5x2 � x � 54x5 � x4 � 2x3 � 8x2 � 2x � 7

x5 � 2x3 � 4x2 � 8x � 374 y2 � 4y

5a2 � 6�11

3�431.5 � 1041.296 � 1022

1.75 � 1017

3.9 � 109y�8153

324x10162x10�24x21�216x21

a3b3

c316a12b4�27y6

x61x533

�4a�11x9165

x4 � x3 � x2 � x � 1

9413�28x8400 � 4a21

2 x2 �12 y2

axnDW

578.6 � 10�7DW

�3.753x2 � 7x � 4 �1

2x � 3

5x2 �12 x � 39a8 � 2a4b3 �

19 b6p3 � q 3

11x3y2 � 8x2y � 6x2 � 6x � 6�9xy � 2y2m6 � 2m2n � 2m2n2 � 8n2m � 6m3�y � 9w � 5

�8�72t4 � 11t 2 � 219y4 � 12y3 � 4y2

x2 � 3x � 2815x7 � 40x6 � 50x5 � 10x49x4 � 1612x3 � 23x2 � 13x � 2

49x2 � 14x � 1x2 �76 x �

13

t 2 � 7t � 12�t � 3� �t � 4�w 2 � 3w4w � 6x5 � 3x3 � x2 � 82x2 � 4x�2x5 � 6x4 � 2x3 � 2x2 � 2

x5 � 2x4 � 6x3 � 3x2 � 910x4 � 7x2 � x �

12�2x2 � 3x � 2

x0x2�2�3y7y2�4y54.4676 � 1095.12 � 10�5

2.09 � 1043.28 � 10�5

1y4t�5y3

8x336x89t 8

1a343t 8�3x�14y111

72

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37. [4.6a] 38. [4.6a] 39. [4.5d] 40. [4.6c] 41. [4.7c] 42. [4.7e] 43. [4.7f] 44. [4.8a]

45. [4.8b]

46. [4.3a] 3, 1.5, , , 47. [4.4d] 48. [4.4d] 49. [4.5b], [4.6a] 50. [2.3b], [4.6b, c]

CHAPTER 5


1. 20 2. 7 3. 72 4. 1 5. 6. 7.8. 9. (a) ; (b)10. (a) ; (b)11. 12.13. 14.15. 16.17. 18.19. 20.21. 22.23. 24. Not factorable using factoring by grouping


1. x 3. 5. 2 7. 9. x 11.13. 15. 17.19. 21.23. 25.27.29.31.33. 35.37. 39.41. 43.45. 47.49. 51. 53.55. 56. 57. 27

58. 59. 60.61. 62.63. 64.

x

y

x � y � 3

(3, 0)

(0, �3)

x

y

(0, 4)

(4, 0)

x � y � 4

y2 � 14y � 49y2 � 49y2 � 14y � 49y2 � 12y � 35p � 2A � q

�x�x �145 ��x�x � �24�

DW�2x2 � 9� �x � 4��x2 � 3� �x � 8��5x2 � 1� �x � 1��4p2 � 1� �3p � 4�

�4x2 � 3� �2x � 3��2x2 � 1� �x � 3��x2 � 2� �x � 3��5a3 � 1� �2a � 7�

�x2 � 2� �x � 3�13 x3�5x3 � 4x2 � x � 1�0.8x�2x3 � 3x2 � 4x � 8�2x3�x4 � x3 � 32x2 � 2�x2y2�x3y3 � x2y � xy � 1�

x2�6x2 � 10x � 3�17xy�x4y2 � 2x2y � 3�2�x2 � x � 4�8x2�x2 � 3�

x2�x � 6�2x�x � 3�x�x � 6�x2y217xyx2

�2x2 � 3� �2x � 3��3x2 � 1� �x � 2��3m3 � 2� �m2 � 5�

�2t 2 � 3� �4t � 1��x2 � 3� �x � 7��x2 � 2� �a � b��x2 � 3� �x � 7�

28�3x2 � 2x � 1�7x3�5x4 � 7x3 � 2x2 � 9�

14�3t 3 � 5t 2 � 7t � 1�3x2y�3x2y � 5x � 1�

y2�3y4 � 5y � 2�x�x � 3�2x�x2 � 5x � 4�2x3 � 10x2 � 8x

3�x � 2�3x � 67x34mn2y24x2

�6112V � l 3 � 3l 2 � 2l

t 2 � 4t � 4�t � 2� �t � 2�,28a � 90�5.25�5�3.5

2x2 � 4x � 2 �17

3x � 2

4x2 � 3x � 59x10 � 16y108a2b2 � 6ab � 4b3 � 6ab2 � ab3�5x3y � y3 � xy3 � x2y2 � 1925t 2 � 20t � 4

6x3 � 7x2 � 11x � 348 � 34y � 5y2x14 � 4x8 � 4x6 � 16

A-20

Answers

65. 66.

67. 69.71. Not factorable by grouping


1. (a) , 8, , 7, ; (b) 13, 8, 7; both 7 and 12 arepositive; (c) 2.3. The coefficient of the middle term, , is negative.4. 5. 6. (a) 23, 10, 5, 2;the positive factor has the larger absolute value; (b) ,

, , ; the negative factor has the larger absolutevalue; (c) 7. (a) , , , ; thenegative factor has the larger absolute value; (b) 23, 10, 5, 2;the positive factor has the larger absolute value; (c) 8.9. 10.11. 12. Prime 13.14. 15.16. , or , or

17. , or, or


1.

3.

�x � 3� �x � 4�

�x � 3� �x � 5�

�x � 3� ��x � 6��x � 3� �x � 6��1�x � 3� �x � 6��x � 2��x � 7�

��x � 2� �x � 7��1�x � 2� �x � 7�3x�x � 4�2p�p � q � 3q2�

x�x � 6� �x � 2��t2 � 7� �t 2 � 2�� y � 6� � y � 2��t � 2� �t � 12��a � 2� �a � 12��x � 2� �x � 12�

�2�5�10�23�x � 3� �x � 8��2�5�10

�23�t � 5� �t � 4��x � 5� �x � 3�

�8�x � 9� �x � 4��x � 3� �x � 4�

�7�8�13

�x7 � 1� �x5 � 1��2x3 � 3� �2x2 � 3�

x

y

(0, 6)

(�2, 0)

y � 3x � 6

x

y

(3, 0)

(0, �5)

5x � 3y � 15

1, 15 16

�1, �15 �16

3, 5 8

�3, �5 �8


1, 12 13

�1, �12 �13

2, 6 8

�2, �6 �8

3, 4 7

�3, �4 �7


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5.

7.

9.

11.

13. 15. 17. Prime19. 21.23. 25.27. 29.31. 33. Prime 35.37. , or , or

39. , or, or

41. 43.45. 47.49. 51. , or

, or 53. 55.57. , or , or

59.61. 63. 65.67. 69. 70.71. 72.73. 74.75. 76. 77. 78. �

72

8327x126x2 � 11xy � 35y2

y3 � 3y2 � 5y16w 2 � 12116w 2 � 88w � 12149w 2 � 84w � 36

28w 2 � 53w � 6616x3 � 48x2 � 8xDW

DW6a8�a � 2� �a � 7��s � 3t� �s � 5t��m � 4n� �m � n��t � 14� ��t � 6�

��t � 14� �t � 6��1�t � 14� �t � 6��p � 5q� �p � 2q�� y � 0.4� � y � 0.2�

�x � 12� ��x � 9��x � 12� �x � 9��1�x � 12� �x � 9��x � 15� �x � 8��a � 12� �a � 11��x � 9� �x � 16�

�x � 24� �x � 3�x2�x � 25� �x � 4��a � 2� ��a � 12��a � 2� �a � 12�

�1�a � 2� �a � 12��x � 10� ��x � 3��x � 10� �x � 3��1�x � 10� �x � 3�

�x � 10�2�x � 6� �x � 7��a2 � 7� �a2 � 5��c2 � 8� �c2 � 7��x � 11� �x � 9�y� y � 9� � y � 5�

x�x � 8� �x � 2��x � 9� �x � 2�� y � 1� � y � 10��d � 2� �d � 5�

�x �13�2

�b � 1� �b � 4�

�x � 2� �x � 7�

�x � 3�2

A-21

Chapters 4–5

79. 29,555 80. 100°, 25°, 55° 81. 15, , 27, , 51, 83. 85.

87. 89.


1. 2.3. 4.5. 6. , or

, or 7. , or , or

8.9.


1. Correct 2. Correct 3. Not correct 4. Not correct5. Not correct 6. Correct 7. Not correct 8. Correct


1. 3.5. 7.9. 11.13. 15.17. 19.21. 23.25. 27.29. , or , or

31.33. 35.37. 39.41. 43.45. 47.49. , or , or

51. , or, or

53. 55.57. 59.61. 63. Prime 65. Prime67. 69.71. 73.

75. 77. 79.

80. 81. 82.

83. 84.85.

86. 87. y-intercept: x-intercept: 88. y-intercept: x-intercept: �16, 0��0, 4�;

�16, 0��0, �4�;y8

x

y

y � Wx � 1

�x �x �8

11��x �x � 4�

q � p � r � 2y �6 � 3x

2x �

y � bm

q �A � 7

pDW6�3x � 4y� �x � y�

�5p � 2q� �7p � 4q��3a � 2b� �3a � 4b��2a � 3b� �3a � 5b��4m � 5n� �3m � 4n�

2x�3x � 5� �x � 1��5t � 8�2�5x2 � 3� �3x2 � 2�3x�8x � 1� �7x � 1�x2�7x � 1� �2x � 3�

�5x � 3� ��3x � 2��5x � 3� �3x � 2��1�5x � 3� �3x � 2��3x � 2� ��3x � 8�

��3x � 2� �3x � 8��1�3x � 2� �3x � 8�p�3p � 4� �4p � 5�5�3x � 1� �x � 2�

�3x � 2� �3x � 8��2x � 1� �x � 1�4�3x � 2� �x � 3��3x � 1� �x � 1�

2�3y � 5� � y � 1�6�5x � 9� �x � 1�4�3x � 2� �x � 3�2�x � 5� ��x � 2�2��x � 5� �x � 2��2�x � 5� �x � 2�

�5x � 11� �7x � 4��24x � 1� �x � 2��3x � 7�2�3x � 2� �3x � 4��7x � 1� �2x � 3��3x � 4� �4x � 5��3x � 1� �x � 2��3x � 2� �3x � 8�

�2x � 1� �x � 1��2x � 3� �2x � 5��3x � 1� �x � 1��3x � 1� �2x � 7�

�5x � 9� �x � 2��2x � 1� �x � 4�

3�2x � 3y� �x � y��2a � b� �3a � b�2�3x � 4� ��x � 1�2��3x � 4� �x � 1��2�3x � 4� �x � 1�

��2x � 1� �3x � 2��2x � 1� ��3x � 2��1�2x � 1� �3x � 2��2x � 1� �3x � 2�

2�5x � 4� �2x � 3��3x � 4� �x � 5��4x � 1� �3x � 5��2x � 5� �x � 3�

2x2�4 � ��bn � 5� �bn � 2��x � 5� �x �

57��x �

14� �x �

34��51

�27�15

1, 9 10

�1, �9 �10

3, 3 6

�3, �3 �6


�1, 14 13

1, �14 �13

�2, 7 5

2, �7 �5


1, 4 5

�1, �4 �5

2, 2 4

�2, �2 �4


1,

�1, � 109�

19

109

19

� 23�

13�

13 ,

23

13

13 ,


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89. y-intercept: x-intercept: 90. y-intercept: x-intercept:


92. y-intercept: x-intercept: 93. 95.97.–105. Left to the student

Margin Exercises, Section 5.4, p. 337

1. 2.3. 4.


1. 3.5. 7.9. 11.13. 15.17. 19.21. 23.25. 27.29. 31.33. 35.37. 39.41. , or , or

43. , or, or

45. 47.49. 51.53. 55.57. 59.61. 63. , or

, or 65. , or , or

67.69. 71.73. 75. 77. Prime79. Prime 81.83. 85.87. 89.91. , or , or

93. 95.97. 98. 99.100. 101. 102.103. 104.105. About 6369 km, or 3949 mi 106. 40°107. 109.111.–119. Left to the student


1. Yes 2. No 3. No 4. Yes 5. No 6. Yes7. No 8. Yes 9. 10.11. 12. 13.14. 15. 16.17. 18. Yes 19. No 20. No 21. No22. Yes 23. Yes 24. Yes 25.26. 27.28. 29. 5�1 � 2t3� �1 � 2t 3�x4�8 � 5x� �8 � 5x�

�a � 5b� �a � 5b�4�t � 4� �t � 4��x � 3� �x � 3�

�3a � 5b�2z3�2z � 5�2� p2 � 9�23�4m � 5�2

�7 � 4y�2�5x � 7�2�t � 2�2�x � 1�2�x � 1�2

�4x5 � 1�2�3x5 � 2�2

�x �x �7717��x �x �

267 �

�x �x � 17��x �x �203 ��x �x � 2�

�x �x � 8��x �x � 217��x �x � �100�DWx3�5x � 11� �7x � 4�6x�x � 5� ��x � 2�

6x��x � 5� �x � 2��6x�x � 5� �x � 2�6�3x � 4y� �x � y��5p � 2q� �7p � 4q��3a � 2b� �3a � 4b��2a � 3b� �3a � 5b�

�4m � 5n� �3m � 4n�2x�3x � 5� �x � 1��5t � 8�2

�5x2 � 3� �3x2 � 2�3x�8x � 1� �7x � 1�x2�7x � 1� �2x � 3�3�2t � 1� ��t � 5�3��2t � 1� �t � 5��3�2t � 1� �t � 5�

�5x � 4� ��x � 1��5x � 4� �x � 1��1�5x � 4� �x � 1�p�3p � 4� �4p � 5�

5�3x � 1� �x � 2��3x � 2� �3x � 8��2x � 1� �x � 1�4�3x � 2� �x � 3��3x � 1� �x � 1�2�3y � 5� � y � 1�6�5x � 9� �x � 1�4�3x � 2� �x � 3�

2�x � 5� ��x � 2�2��x � 5� �x � 2��2�x � 5� �x � 2��3a � 1� ��3a � 5�

��3a � 1� �3a � 5��1�3a � 1� �3a � 5��24x � 1� �x � 2��3x � 7�2

�3x � 2� �3x � 4��7x � 1� �2x � 3��3x � 4� �4x � 5��3x � 1� �x � 2��3x � 2� �3x � 8��2x � 1� �x � 1��2x � 3� �2x � 5��3x � 1� �x � 1��2x � 7� �3x � 1��3x � 5� �x � 3�

�2x � 1� �x � 4��2x2 � 5� �x2 � 3��2x � 3� �2x � 3��5x � 3� �7x � 8�

�x � 4� �3x � 4��2x � 3� �3x � 2��x � 1� �x � 4��x � 7� �x � 2�

2�5x � 4� �2x � 3�3�2x � 3� �x � 1��4x � 1� �3x � 5��2x � 1� �3x � 2�

�x3a � 1� �3x3a � 1��2xn � 1� �10xn � 3��5

2 , 0��0, �5�;�4

5 , 0��0, 4�;�5

8 , 0��0, 23�;

�6.5, 0��0, �5�;

A-22

Answers

30.31.32.


1. Yes 3. No 5. No 7. No 9.11. 13. 15. 17.19. 21. 23.25. 27. 29.31. 33. 35.37. 39. 41. Yes 43. No45. No 47. Yes 49.51. 53.55. 57.59. 61.63. 65.67. 69.71. 73.75. 77.79.81. 83.85. 87.89. 91. 92. 400 93. 94.95. 2 96. 97. 98.99. 100. 101.

102. 103. Prime

105. 107.109.111. 113.115. 117.

119. 121.

123. 125.127. 129. 9 131. Not correct133. Not correct


1. 2.3. 4. �2y � z� �4y2 � 2yz � z2��3x � y� �9x2 � 3xy � y2�

�4 � y� �16 � 4y � y2��x � 2� �x2 � 2x � 4�

� y � 4�2�3bn � 2�2�9 � b2k� �3 � bk� �3 � bk�

�x �1x ��x �

1x �x�x � 6�

�0.8x � 1.1� �0.8x � 1.1�p�0.7 � p� �0.7 � p�2x�3x �

25� �3x �

25�3x3�x � 2� �x � 2�

�x4 � 24� �x2 � 22� �x � 2� �x � 2�2x�3x � 1�2�x � 11�2

x

y

(5, 0)

(0, �3)

3x � 5y � 15

x

y

(0, 6)

(�1, 0)

y � 6x � 6

25a4b6y12

12�x2 � 2xyx2 � 4xy � 4y2�160

�0.9�56�11DW

�4m2 � t 2� �2m � t� �2m � t��5 �17 x� �5 �

17 x�

� y �14� � y �

14��x6 � 4� �x3 � 2� �x3 � 2�

�1 � y4� �1 � y2� �1 � y� �1 � y�5�x2 � 9� �x � 3� �x � 3��a2 � 4� �a � 2� �a � 2�

�7a2 � 9� �7a2 � 9��0.3y � 0.02� �0.3y � 0.02��1

4 � 7x4� �14 � 7x4�x�6 � 7x� �6 � 7x�

2�2x � 7� �2x � 7��2x � 5y� �2x � 5y��4a � 3� �4a � 3��10 � k� �10 � k�

�5t � m� �5t � m��a � b� �a � b��t � 7� �t � 7�� p � 3� � p � 3�

� y � 2� � y � 2�4�3a � 4b�2�9a � b�2

�a � 3b�2�2p � 3q�2�1 � 2x2�25� y2 � 1�2�7 � 3x�23�2q � 3�2

x�x � 9�22�x � 1�2�4y � 7�2�q2 � 3�2�x � 2�2�x � 1�2�x � 8�2

�x � 7�2

�7p2 � 5q3� �7p2 � 5q3��4 �

19 y4� �2 �

13 y2� �2 �

13 y2�

�9x2 � 1� �3x � 1� �3x � 1�

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5.6.7.8.


1. 3.5. 7.9. 11.13. 15.17. 19.21. 23.25. 27.29.31.33.35.

37. 39. 41.

42. 43. 44. 45.

46. 47.49.51. 53.55.


1. 2.3. 4.5. 6.7. 8.9. 10. 11.12.13.


1. 3. 5.7. 9.11. 13.15. Prime 17. 19.21. , or , or

23. Prime25.27. 29.31. 33. 35.37. 39.41. 43.45. 47. , or 49. 51. 53.55. 57.59. 61.63. 65.67.69.71.73. �1 � 4x6y6� �1 � 2x3y3� �1 � 2x3y3�

�4 � p2q2� �2 � pq� �2 � pq�7�x � y� �x2 � xy � y2� �x � y� �x2 � xy � y2��a �

15b� �a �

15b�

a3�a � b� �a � 5b�r3�rs � 2� �rs � 8��mn � 8� �mn � 4��m � 20n� �m � 18n�

�a � b� �a � 2b��12 a �

13 b�2

� y2 � 5z2�2�7m2 � 8n�2�4x � 3y�2�a � 2b�2�2b � a�2�3q � p� �2q � 1�

�n � p� �n � 2��x � 1� �x � 1 � y��a � b� �2x � 1�2�r�h � r�

9xy�xy � 4�m�x2 � y2�19�1

3 x3 � 4�2x3�x � 3� �x � 1��1 � y4� �1 � y2� �1 � y� �1 � y�

4�x2 � 4� �x � 2� �x � 2�2�x � 2� ��x � 5�

2��x � 2� �x � 5��2�x � 2� �x � 5�x3�x � 7�2x �x � 3� �x2 � 7�

3x�3x � 5� �x � 3�3�4x � 1� �4x � 1��x � 2� �x � 2� �x � 3�x�x � 12�2

�2x � 3� �x � 4��a � 5�23�x � 8� �x � 8�

15�a � 2b� �a2 � 2ab � 4b2��p2 � 9q2� �p � 3q� �p � 3q�

�xy � 1� �xy � 4��x2 � y2�2�a � b� �x2 � y��a � b� �2x � 5 � y2�2p4q2�5p2 � 2pq � q2�

5�3x2 � 2y� �x2 � y�8x�x � 5� �x � 5��3x2 � 2� �x � 4�2x2�x � 1� �x � 3�

�x3 � 4�23�m2 � 1� �m � 1� �m � 1�

4�3a2 � 4�y �3x2 � 3xy � y2�1

3 �1

2 xy � z� �1

4 x2y2 �

12 xyz � z2�

3�xa � 2yb� �x2a � 2xayb � 4y2b��x2a � yb� �x4a � x2ayb � y2b�x2 � 0.04x � 0.05

w 2 �23 w �

194y10 � 9

16x6a8b18

343y15

DW�t 2 � 4y2� �t 4 � 4t 2y2 � 16y4�

�z � 1� �z2 � z � 1� �z � 1� �z2 � z � 1�2y � y � 4� � y2 � 4y � 16�8�2x2 � t 2� �4x4 � 2x2t 2 � t 4��x � 0.1� �x2 � 0.1x � 0.01�

5�x � 2z� �x2 � 2xz � 4z2�r �s � 4� �s2 � 4s � 16�3�2a � 1� �4a2 � 2a � 1�2� y � 4� � y2 � 4y � 16��a �

12�

�a2 �

12 a �

14��a � b� �a2 � ab � b2�

�2x � 3� �4x2 � 6x � 9��4y � 1� �16y2 � 4y � 1��2 � 3b� �4 � 6b � 9b2�� y � 2� � y2 � 2y � 4��2a � 1� �4a2 � 2a � 1�� y � 5� � y2 � 5y � 25�

�x � 1� �x2 � x � 1��z � 3� �z2 � 3z � 9�

�x � 0.3� �x2 � 0.3x � 0.09��3x � 2y� �9x2 � 6xy � 4y2� �3x � 2y� �9x2 � 6xy � 4y2�2xy �2x2 � 3y2� �4x4 � 6x2y2 � 9y4��m � n� �m2 � mn � n2� �m � n� �m2 � mn � n2�

A-23

Chapter 5

75. 77. 79.

81. 82.83. 84.

85. 86. 78 87. 88.

89. 91.93. 95.97. 99.101. 103.


1. 3, 2. 7, 3 3. 4. 0, 5. , 3

6. , 7 7. 3 8. 0, 4 9. 10. 3, 11. , 2

12. , 3 13. 14. 0, 3




1. 3. , 8 5. , 11 7. 0, 9. 0, 11. 13. , 3 15. 4, 17. 0, 19.21. 23. 0, 25. 27. , 2

29. 3, 5 31. 0, 8 33. 0, 35. , 4 37.39. 41. 4 43. 0, 45. 47.49. 51. 53. , 9 55.57. 59. 61.63. , 4 65. , 3 67. 69.70. 71.72. 73. 74.75. , 4 77. , 9 79. 81. , 4

83. Answers may vary. (a) ;(b) ; (c) ;(d) ; (e) 85. 2.33, 6.7787. 0, 2.74


1. Length: 24 in.; width: 12 in. 2. Height: 25 ft; width: 10 ft3. (a) 342 games; (b) 9 teams 4. 22 and 23 5. 24 ft6. 3 m, 4 m


1. O 2. M 3. K 4. I 5. G 6. E 7. C 8. A9. H 10. B


1. Length: 6 cm; width: 4 cm 3. Length: 12 ft; width: 2 ft5. Height: 4 cm; base: 14 cm 7. Base: 8 m; height: 16 m9. 182 games 11. 12 teams 13. 4950 handshakes15. 25 people 17. 20 people 19. 14 and 1521. 12 and 14; and 23. 15 and 17; and �17�15�14�12

40x3 � 14x2 � x � 0x2 � 25 � 04x2 � 4x � 1 � 0x2 � 7x � 12 � 0

x2 � x � 12 � 0

�4�18, 18�3�5

310�

103�4.5�16a2 � b2

�a � b�2DW�1�1��3, 0�, �5, 0��

52, 0�, �2, 0��4, 0�, �1, 0�

45, 32�2�

710, 7

10�1, 23

�14, 23�1, 53

65�3

�23, 23�4�18

�9�5, �123, 12

13, �20

�1

10, 127

23

14�

15�

52, �4

�18�3�12�3�4, �9

��5, 0�, �1, 0��3

�572�

43, 43�4

�2173�

14, 23�4

� y � 4 � x�2� y � 1�3�x � 2� �x � 2� �x � 1��a2 � 1� �a � 4�

� y � 3� � y � 3� � y � 2��5x � 4� �x � 1.8��3.5x � 1�2�a � 1�2�a � 1�2

�x �x � 32�X �A � 7a � b

�1411

6x2 � 11xy � 35y29x2 � 30xy � 25y29x2 � 30xy � 25y29x2 � 25y2

DW�7x � 8y�2�q � 1� �q � 1� �q � 8�

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-477

42-6


25. Hypotenuse: 17 ft; leg: 15 ft 27. 32 ft 29. 9 ft31. Dining room: 12 ft by 12 ft; kitchen: 12 ft by 10 ft33. 4 sec 35. 5 and 7 37. 39. Factor40. Factor 41. Product 42. Common factor43. Trinomial 44. Quotient rule 45. y-intercept46. Slope 47. 35 ft 49. 5 ft 51. 30 cm by 15 cm53. 7 ft


1. True 2. False 3. True 4. False 5. True


1. 2. 12x 3. 4.5. 6. 7.8. 9.10. 11.12. 13.14. 15.16. 17. 18. Prime19. 20.21. 22. 23.24. 25.26. 27. 28.29. 30. 31.32. 33.34. 1, 35. , 5 36. 37. , 1 38.39. , 8 40. Height: 6 cm; base: 5 cm 41. and

; 16 and 18 42. and ; 17 and 19 43. 3 ft44. 6 km 45. 46.47. Answers may vary. The area of a rectangle is The length is 1 m greater than the width. Find the lengthand the width. 48. Because Sheri did not first factorout the largest common factor, 4, her factorization will notbe “complete” until she removes a common factor of 2 fromeach binomial. 49. 2.5 cm 50. 0, 2 51. Length: 12; width: 6 52. No solution 53. 2, , 54.55.


1. [5.1a] 2. [5.2a] 3. [5.5b] 4. [5.1b] 5. [5.1c] 6. [5.1b] 7. [5.2a] 8. [5.3a], [5.4a] 9. [5.5d] 10. [5.2a] 11. [5.3a], [5.4a] 12. [5.5d] 13. [5.5b] 14. [5.5d] 15. [5.5b] 16. [5.3a], [5.4a] 17. [5.1c] 18. [5.5d] 19. [5.3a], [5.4a] 20. [5.3a], [5.4a] 21. [5.2a] 22. [5.6a] 23. [5.8b] , 5 24. [5.8b] 25. [5.8b] 26. [5.9a] Length: 8 m; width: 6 m�4, 7

�5, 32�4�10a � 3b� �100a2 � 30ab � 9b2�3�m � 2n� �m � 5n�

3t�2t � 5� �t � 1��2x � 3� �2x � 5�

5�4 � x2� �2 � x� �2 � x��x3 � 3� �x � 2�

�5x � 1� �x � 5��7x � 6�23�x2 � 4� �x � 2� �x � 2�

5�3x � 2�23�w � 5� �w � 5�3m�2m � 1� �m � 1�

�x � 4� �x � 3��2x � 3� �2x � 3�2�5x � 6� �x � 4�

x�x � 3� �x � 1�x�x � 5��x2 � 2� �x � 1�2y2�2y2 � 4y � 3�

�x � 5�2�x � 5� �x � 2�4x3

�� 2�x2�2, 54, 35

2�3

DW

90 m2.DW��

32, 0�, �5, 0��5, 0�, ��4, 0�

�17�19�16�18�2

�4, 3223�4, 3�7�3

32�x2 � 2y2z2� �x2 � 2y2z2��m � t� �m � 5�3�2a � 7b�2�xy � 4� �xy � 3��7b5 � 2a4�2�5x � 2�2�x � 5� �x � 3�3�x � 3� �x � 3�

2�3x � 1�2�2x � 1� �x � 4��x � 3�22�3x � 4� �x � 6��3x � 5�2

�2x � 5� �4x2 � 10x � 25�x�x � 6� �x � 5�3�2x � 5�24x4�2x2 � 8x � 1�

�4x2 � 1� �2x � 1� �2x � 1��x3 � 2� �x � 4�2�x � 5� �x � 5�3x�3x � 5� �x � 3�

�x2 � 9� �x � 3� �x � 3��3x � 1� �2x � 1��x2 � 3� �x � 1�3x�2x2 � 4x � 1�

�x � 7�2�x � 6� �x � 2��3x � 2� �3x � 2�x�x � 3�5�1 � 2x3� �1 � 2x3�5y2

DW

A-24

Answers

27. [5.9a] Height: 4 cm; base: 14 cm 28. [5.9b] 5 ft29. [5.8b] 30. [5.8b] 31. [5.9a] Length: 15; width: 3 32. [5.2a] 33. [5.8b] 34. [4.6b], [5.5d] (d)

CHAPTER 6


1. 3 2. �8, 3 3. None 4.

5. 6. 7. 5 8.

9. 10. 11. 12.

13. �1 14. �1 15. �1 16. 17.


1. Correct 2. Correct 3. Not correct 4. Not correct5. Not correct 6. Not correct 7. Correct 8. Correct


1. 0 3. 8 5. 7. �4, 7 9. �5, 5 11. None

13. 15. 17.

19. 21. 23. 25.

27. 29. 31. 33.

35. 37. 39. 41.

43. 45. 47. �1 49. �1 51. �6

53. 55. 57. 59.

61. 63. 65.

67. 69. 71. 73. 18 and 20;

�18 and �20 74. 3.125 L 75.76. 77.78. 79.80. 81. 82. Prime83. 84. 85.

87. 89.x � y

x � 5y�t � 9�2�t � 1��t 2 � 9� �t � 1�

x � 2y�a � 7b� �a � 2b��4x � 5y�2�x � 7� �x � 2�10�x � 7� �x � 1��2 � t� �2 � t� �4 � t2��2y2 � 1� � y � 5�

x3�x � 7� �x � 5��a � 8�2�x � 8� �x � 7�

DW5�a � 6�

a � 1x � 4x � 2

�t � 2� �t � 2��t � 1� �t � 1�

2aa � 2

�a � 3� �a � 3�a�a � 4�

x � 2x � 2

2dc2

56x3

�x � 1

t � 2t � 2

t � 22�t � 4�

6t � 3

32

x2 � 1x � 1

a � 1

x � 5x � 5

a � 3a � 4

a � 3a � 2

m � 12m � 3

x � 3x

8p2q3

x2

4� y � 6� � y � 7�� y � 6� � y � 2�

�1�3 � x��1�4 � x�

2x�x � 1�2x�x � 4�

�4x� �3x2��4x� �5y�

�52

x � 52

a � 2a � 3

y � 24

x � 2x � 1

2x � 12x � 13x � 2

x4

�x � 8� ��1��x � y� ��1�

�x � 1� �x � 2��x � 2� �x � 2�

�2x � 1�x�3x � 2�x

�83, 0, 25

�a � 4� �a � 8��2

3, 0�, �1, 0��5, 0�, �7, 0�

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91.

You get the same number you selected. To do a numbertrick, ask someone to select a number and then performthese operations. The person will probably be surprisedthat the result is the original number.


1. 2. 3. 4. 5.

6. 7. 8. 9.

10. 11.


1. 3. 5. 7. 9.

11. 13. 15. 17.

19. 21. 23. 25. 27.

29. 31. 33. 35. 37.

39. 40. Height: 7 in.; base: 10 in.41. 42.

43. 44. 45. 46. 47.

49.


1. 144 2. 12 3. 10 4. 120 5. 6. 7.8. 9. 10.11. 12.


1. 108 3. 72 5. 126 7. 360 9. 500 11.13. 15. 17. 19.21. 23.25. 27.29. 31.33. 35.37. 39.41. 43. 44.45. 46. 47.48. 49.50. 51.52. 53.54. 55. 24 mina15; a5; a20

120x3; 2x2; 240x548ab3; 4ab; 192a2b420x2; 10x; 200x348x6; 16x5; 768x11

120x4; 8x3; 960x7�x � 7� �x � 3��x � 3�2�x � 7� �x � 3��x � 3� �x � 3�

2x�3x � 2��x � 3�2DW6x3�x � 2�2�x � 2�18x3�x � 2�2�x � 1�

10v�v � 4� �v � 3��2 � 3x� �2 � 3x��m � 3� �m � 2�2�a � 1� �a � 1�2

t�t � 2�2�t � 4��x � 2� �x � 2� �x � 3�t�t � 2� �t � 2�6� y � 3�

18x2y212x323180

29120

6572

3x�x � 1�2�x � 1�7�t2 � 16� �t � 2�� y � 1�2� y � 4�60x3y29

40

1110

14

35144

a � 15ab2�a2 � 4�

�1

b21

a15b204x6

y10125x18

y124y8

x6

�2p2 � 4pq � 4q28x3 � 11x2 � 3x � 12�x � x � 77�

DWx � 1x � 1

y � 32y � 1

c � 1c � 1

32

�x � 2�2

xa � 5

3�a � 1�154

158

12

�x � 1�2

x�a � 2� �a � 3��a � 3� �a � 1�

ba

14

310

x2 � 4x � 7x2 � 2x � 5

a � b1

x2 � y2x4

y � 1y � 1

�x � 3� �x � 2�x � 2

x � 3x � 2

3a2

a � 5�x � 3� �x � 2��x � 5� �x � 5�

58

67

x2 � 31

x � 52x3 � 1x2 � 5

27

� x

�10x10

5�2x � 5� � 25

10�

10x � 25 � 2510

A-25

Chapters 5–6


1. 2. 3. 4. 5.

6. 7.

8. 9. 10.

11.


1. 1 3. 5. 7. 9.

11. 13. 15.

17. 19. 21.

23. 25. 27.

29. 31.

33. 35. 37.

39. 41. 43. 45.

47. 49. 51. �1 53.

55. 57.

59. 61. 63.

64. 65. 66.

67. 68. 69.

70. 71.

x

y

y � 3

x

y

2y � x � 10 � 0

x

y

y � qx � 5

25x4y6

1x12y21

x6

25y21

8x12y913y3 � 14y2 � 12y � 73

x2 � 1DW5t � 12

�t � 3� �t � 3� �t � 2�

a2 � 7a � 1�a � 5� �a � 5�

2x � 6y�x � y� �x � y�

�x2 � 9x � 14�x � 3� �x � 3�

5x � 2x � 5

a � b

2b � 14b2 � 16

y � 3�x � 7x � 6

�1t

14

3a � 2�a � 1� �a � 1�

2x2 � 4x � 34�x � 5� �x � 3�

7a � 6�a � 2� �a � 1� �a � 3�

2x2 � 8x � 16x�x � 4�

11a10�a � 2�

3x � 1�x � 1�2

6z � 4

x2 � 6x�x � 4� �x � 4�

11x � 23x�x � 1�

6x�x � 2� �x � 2�

x2 � 4xy � y2

x2y24 � 3t

18t 34x � 6y

x2y2

4124r

2x � 5x2

2x � 3x � 5

63 � x

�2x � 113�x � 4� �x � 4�

x � 1x � 3

x � 54

8x � 88�x � 16� �x � 1� �x � 8�

2x2 � 16x � 5�x � 3� �x � 8�

4x2 � x � 3x�x � 1� �x � 1�2

9x � 1048x2

10x2 � 9x48

6x � 4x � 1

3 � xx � 2

79

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42-6


72. 73. �8 74.

75. 3, 5 76. �2, 9

77. Perimeter:

area:

79.

81.

83.–85. Left to the student


1. 2. 3. 4.

5. 6. 7.

8. 9. 10.


1. 3. 1 5. 7. 9.

11. 13. 15.

17. 19. 21.

23. 25. 27. 29. 31.

33. 35. 37.

39. 41. 43. 0 45.

47. 49. 51. 53.

55. 57. 58. 59. 60.

61. 62. 63. 64.

65. 67.

69. Missing side: area:



1. 2. 3. 4. 5.

6.x

x � 1

7x2

3�2 � x2�136

5x

x � 17x2

3�2 � x2�136

5

�2a3 � 15a2 � 12a � 902�a � 6�2

�2a � 15a � 6

;

x2 � xy � x3 � x2y � xy2 � y3

�x2 � y2� �x � y�2�x � y�30

�x � 3� �x � 4�

�4 � ��r 2x2 � 9x � 1810x3

6x3

18x3b20

a830x12x5DW

2x � y

z � 32z � 1

2a � 32 � a

202y � 1

�92x � 3

18x � 5x � 1

x � 3�x � 3� �x � 1�

12

9x � 12�x � 3� �x � 3�

2x � 4x � 9

4a2 � 25

x � 2x � 7

8y � 1

13a

83

�2a2

�x � a� �x � a�y � 19

4y2s � st � s2

�t � s� �t � s�

3 � 5t2t�t � 1�

2x � 40�x � 5� �x � 5�

4x2 � 13xt � 9t2

3x2t 2

7z � 1212z

�a � 410

1x � 1

4x

6x2 � 2x � 23x�x � 1�

x � 13�x � 3� �x � 3�

�8y � 28� y � 4� � y � 4�

4x � 3x � 2

3x � 13

x2 � 48�x � 7� �x � 8� �x � 6�

�x � 715x

x2 � 2x � 12x � 1

5y

411

11z4 � 22z2 � 6�z2 � 2� �z2 � 2� �2z2 � 3�

�z � 6� �2z � 3��z � 2� �z � 2�

y2 � 2y � 815

16y � 2815

;

56

x

y

x � �5

A-26

Answers


1. 3. 5. �6 7. 9. 11. 8

13. 15. 17. 19.

21. 23. 25.

27. 29. 1 31. 33.

35. 36. 0 37.38. 39. 40.

41. 14 yd 42. 12 ft, 5 ft 43.

45.


1. 2. 3. 3 4. 5. 1 6. 2 7. 4



Study Tip, p. 441

1. Rational expression 2. Solutions 3. Rationalexpression 4. Rational expression 5. Rationalexpression 6. Solutions 7. Rational expression8. Solutions 9. Solutions 10. Solutions11. Rational expression 12. Solutions 13. Rationalexpression


1. 3. 5. 7. �6 9. 11. �4, �1

13. �4, 4 15. 3 17. 19. 5 21. 5 23.25. �2 27. 29. 31. No solution 33. �5

35. 37. 39. No solution 41. No solution43. 4 45. No solution 47. 49. 751. 53. Quotient 54. Product 55. Reciprocals56. Factoring 57. Greatest 58. Not 59. Subtract60. Additive inverses 61. 63. Left to the student


1. hr 2. Greg: 40 mph; Nancy: 60 mph 3.4. 0.28 5. 6. 7. About34.6 gal 8. 90 whales 9. Yes; approximately 224 walks10. 24.75 ft 11. 7.5 ft


1. K 2. E 3. C 4. N 5. D 6. O 7. F 8. H9. B 10. A

2.4 fish�yd2124 km�h58 km�L3 3

7

� 16

DW�2, 2

12

53

172�

132

52

143

247

472

4029

65

�18

32

332

5x � 33x � 2

�x � 1� �3x � 2�5x � 3

5� p � 2� � p � 10�50� p2 � 2�� p � 5�2� p � 5�24x4 � 3x3 � 2x � 7

DW4x � 15x � 3

acbd

60 � 15a3

126a2 � 28a32a2 � 4a5 � 3a2

p2 � q2

q � p

abb � a

�1a

yy � 1

x � 8

2x � 1x

1 � 3x1 � 5x

13

254

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1. hr 3. min 5. hr 7. min

9. min 11. Sarah: Rick: 13. Passenger: 80 mph; freight: 66 mph 15. 20 mph17. Hank: Kelly: 19. Ralph: Bonnie: 21. 3 hr 23.25. 27. 66 g 29. 1.92 g 31. 1.75 lb33. 35. (a) 0.332; (b) 243 hits; (c) 232 hits

37. 22 in.; 55.8 cm 39. 57.9 cm 41.43. 287 trout 45. 200 duds 47. (a) 4.8 tons; (b) 48 lb49. 51. 53. 55. 15 ft 57.

59. 60. x 61. 62.

63. 64.

65. 66.

67. 68.

69. Ann: 6 hr; Betty: 12 hr 71. 73.


1. 2. 0.7; 3. 50 volts

4. 1,575,000 tons 5. 0.6; 6. 16 hr

7. 8. 9. 10.

11. 490 m 12. 1 ohm

y �5xz2

wy � 1

2 xzy �9

x2y � 7x2

y �0.6x

y � 0.7x25 ; y � 2

5 x

t �ab

b � a27 3

11 min

y

x

y � �x � 425

y

x

y � ��x � 234

y

x

x � 3y � 6

y

x

3x � 2y � 12

y

x

y � �2x � 6

y

x

y � 2x � 6

1x

1x11x11

DW353

83

212

7 12 ; 23 3

5 in.7 14 ;

1 1139 kg

2.3 km�h

59 divorce�marriage8 km�h

5 km�h;19 km�h14 km�h;

70 km�h30 km�h;3 34

22 293 15

1625 572 2

9

A-27

Chapter 6


1. 3. 5.7. 285,360,000 cans 9. 11. 90 g 13. 40 kg

15. 98; 17. 36; 19. 0.05;

21. 3.5 hr 23. ampere 25. 1.92 ft 27.

29. 31. 33.

35. 37. 39. 36 mph 41. 2.5 m

43. 50 earned runs 45. 729 gal 47.49. 50. 51.52. 53.54. 55.56. Not factorable 57.58. 59.

60. 61.

63. Q varies directly as the square of p and inversely as thecube of q. 65. $7.20


1. False 2. True 3. True 4. True 5. False


1. 0 2. 6 3. �6, 6 4. �6, 5 5. �2 6. None

7. 8. 9. 10.

11. 12. 13. 14.

15. 16. 17.

18. �1 19. 20. 21. 22.

23. 24. 25.

26. 27. 28. 8 29. �5, 3 30.

31. 32. 95 mph, 175 mph33. 160 defective calculators 34. (a) (b) (c)35. 10,000 blue whales 36. 6 37.

38. 39. 40. 41. 20 min

42. About 77.7 43. 500 watts

44. used to find an equivalent

expression for each rational expression with the LCM as the

least common denominator 45.

used to find an equivalent expression for each rationalexpression with the LCM as the least common denominator

3x � 10�x � 2� �x � 2�

;DW

5x � 6�x � 2� �x � 2�

;DW

y �2xz

y � 6xzy �2500

x

y � 4x

9 13 c4 1

5 c;1213 c;

240 km�h, 280 km�h

5 17 hrc � d

z1 � z

2�x � 2�x � 2

�x2 � x � 26�x � 5� �x � 5� �x � 1�

2x � 3x � 2

x � 52x

4x � 4

d � c2a

a � 1

�3x � 18x � 7

� y � 2� � y � 2� � y � 1�4�a � 2�

30x2y22x2 � 2xx � 1

�20t6

2t � 1

a � 65

y � 5y � 5

7x � 3x � 3

x � 2x � 1

�

4�w � t� �w 2 � wt � t 2� �w � t� �w 2 � wt � t 2�

3�x � y� �x2 � xy � y2��a � 7b� �a � 2b��4x � 5y�2

�x � 7� �x � 2�10�x � 7� �x � 1��2 � t� �2 � t� �4 � t 2��2y2 � 1� � y � 5�

x3�x � 7� �x � 5��a � 8�2�x � 8� �x � 7�DW

y �xz

5wpy � 3

10 xz2

y � xzy �0.0015

x2y � 15x2

160 cm329

y �0.05

xy �

36x

y �98x

66 23 cm

94 ; y � 9

4 x215 ; y � 2

15 x5; y � 5x

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42-6


46. 4; used to clear fractions 47.

method 1: used to multiply by 1 using method 2:LCM of the denominators in the numerator used to subtractin the numerator and LCM of the denominators in the denominator used to add in the denominator

48. 49.

50. They are equivalent proportions.


1. [6.1a] 0 2. [6.1a] �8 3. [6.1a] �7, 7 4. [6.1a] 1, 2

5. [6.1a] 1 6. [6.1a] None 7. [6.1c]

8. [6.1d] 9. [6.2b]

10. [6.3c] 11. [6.4a]

12. [6.5a] 13. [6.4a] 14. [6.5a]

15. [6.4a] 16. [6.5a]

17. [6.5b] 18. [6.6a]

19. [6.7a] 12 20. [6.7a] �3, 5 21. [6.8b] 16 defectivespark plugs 22. [6.8b] 50 zebras 23. [6.8a] 12 min24. [6.8a] Craig: Marilyn: 25. [6.8b] 15

26. [6.9b] $495 27. [6.9e] 28. [6.9c]

29. [6.8a] Rema: 4 hr; Reggie: 10 hr 30. [6.6a]


1. [1.2e] 3.5 2. [4.3d] 3. [4.3g] 3, 2, 1, 0; 3 4. [4.3i] None of these5. [2.5a] $16.74 6. [6.8b] 27 lb 7. [6.8a] 30 min8. [6.9b] (a) Let muscle weight, in pounds, and body weight, in pounds; ; (b) 76.8 lb9. [2.5a] $2500 10. [6.8a] 35 mph, 25 mph11. [5.9a] 14 ft 12. [2.6a] 34 and 3513. [4.3e] 14. [1.8c]

15. [4.1e], [4.2a, b] 16. [6.6a]

17. [4.7e] 18. [4.4a]

19. [6.1d] 20. [6.2b] 2 21. [6.4a]

22. [6.5a] 23. [4.6a]

24. [4.6c] 25. [4.6b]

26. [4.8b]

27. [5.3a], [5.4a] 28. [5.5b] 29. [5.5d] 30. [5.6a] 31. [2.3c] 332. [5.8b] 33. [5.8b] �5, 4�4, 12

�w � 1� �w 2 � w � 1��7x � 1� �7x � 1�

�3x � 5y�2�9a � 2� �a � 6�

2x2 � 7x � 7 ��3

x � 1

4x6 � 136x2 � 60x � 25

a2 � 92x � 6

�x � 2� �x � 2�

x � 42

3� y � 2�

2x5 � 6x4 � 2x3 � 10x2 � 3x � 9�2xy2 � 4x2y2 � xy3

8x � 1217x

94x8

38 x � 12x3 � 3x2 � 2

M � 0.4BB �M �

1, �2, 1, �1

3a � 22a � 1

y �250

xQ � 2.5xy

45 km�h65 km�h;

3y � 1y

x2 � 2x � 7�x � 1�2�x � 1�

�x2 � 7x � 15�x � 4� �x � 4� �x � 1�

8t � 3t�t � 1�

2x � 5x � 3

�3x � 3

8 � 2tt 2 � 1

23 � 3xx3� y � 3� � y � 3� � y � 7�

�5x � 1��x � 1�3x�x � 2�

a � 52

3x � 7x � 3

10a�a � b� �b � c�

5�a � 3�2

a

LCM�LCM;

4�x � 2�x�x � 4�

;DWDW

A-28

Answers

34. [2.7e] 35. [5.8a] 0, 436. [5.8b] 0, 10 37. [5.8b]

38. [2.4b] 39. [6.7a] 2

40. [6.7a] No solution41. [3.2a] 42. [3.3a]

43. [3.3b] 44. [3.3b]

45. [6.9a] ; 46. [6.9c]

47. [3.4a] Not defined 48. [3.4a] 49. [3.4a] �3,009,500 50. [3.3a] 51. [3.4b] �3,009,500 acres per year 52. [4.4c], [4.6a] 1253. [4.6b, c] 54. [5.5d] 55. [5.8a] 56. [6.1a], [6.6a]

CHAPTER 7


1. Yes 2. No 3. Yes 4. No 5. Yes 6. Yes7. No 8. (a) �4; (b) 148; (c) 31; (d)9. (a) �33; (b) �3; (c) (d)

10. 11.

x

y

g(x) � 5 � x 2

x

y

f (x) � x � 4

6a � 6h � 7�5a � 2;8a2 � 6a � 4

0, 3, 524, �7, 122�a16 � 81b20� �a8 � 9b10� �a4 � 3b5� �a4 � 3b5�

16y6 � y4 � 6y2 � 9

�0, 968,845,000��

37

y �10x

; 225

2503y � 2

3 x

x

y

x � �3y � 1

x

y

3x � 5y � 15

x

y

y � qx

x

y

a �t

x � y

�20, 20�x �x � �26�

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12. 13. Yes 14. No15. No 16. Yes17. About 97 millionsubscribers 18. About155 million subscribers


1. �13.3 2. �14.4 3. 14 4. 34


1. 2.

3. 4.

5. 6.

7. 8.

9.


1. Yes 3. Yes 5. Yes 7. No 9. Yes 11. No13. Yes 15. (a) 9; (b) 12; (c) 2; (d) 5; (e) 7.4; (f ) 5 2

3

y � x 4 � 2

�5

�5

5

15

y � |x � 3|

�10 10

�10

10

y � x 3

�10 10

�10

10

y � 3x 2 � 4x � 1

�5

�5

5

15y � 1 � x 2

�10 10

�10

10

y � x 2 � 2

�10

�5

10

15y � �3x � 4

�10 10

�10

10

y � �2x � 3

�10 10

�10

10

y � x � 4

�10 10

�10

10

x

y

t(x) � 3 � �x�

A-29

Chapters 6–7

17. (a) �21; (b) 15; (c) 2; (d) 0; (e) 18a; (f )19. (a) 7; (b) �17; (c) 6; (d) 4; (e) (f )21. (a) 0; (b) 5; (c) 2; (d) 170; (e) 65; (f )23. (a) 1; (b) 3; (c) 3; (d) 11; (e) ; (f )25. (a) 0; (b) �1; (c) 8; (d) 1000; (e) �125; (f )27. (a) 159.48 cm; (b) 167.73 cm 29. atm; atm;

atm 31. 1.792 cm; 2.8 cm; 11.2 cm

33.

35.

37. 39.

41.

43.

�2 21 4

2

�2

�4

�4�5 �3 �1�1

�3

�5

1

345

3 5

y

x

g (x)� |x � 1|

x

y

f (x) � 2 � �x�

x

y

f (x) � qx � 1

x

y

g(x) � �2x � 3

x

y

f (x) � 3x � 1

�2 21 4

2

�2

�4

�4�5 �3 �1�1

�3

�5

345

3 5

y

x

f (x)� �2x

4 133

1 10111 20

33

�27a3�a � h� � 1�a � 1� � 1

32a2 � 12a3a � 3h � 43a � 2;

3a � 3

�2 4�1 2

0 02 �4

x f �x�

�1 �40 �11 22 5

x f �x�

�3 �1�2 0�1 1

0 21 12 03 �1

x f �x�

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42-6


45.

47.

49.

51.

53. Yes 55. Yes 57. No 59. No 61. About1150 stations 63. About 1.9 billion images 65.67. Quadrants 68. Relation 69. Function; domain; range; domain; range 70. Graph 71. Inputs72. Solutions 73. Addition principle 74. Vertical-linetest 75. 26; 99 77.


1.2. (a) �4; (b) (c) �3, 3; (d)3. Domain: all real numbers; range: all real numbers4. All real numbers 5. is a real number and


1. (a) 3; (b) (c) �2, 0; (d)3. (a) (b) ; (c) (d) � y � 1 � y � 4�2

14 ;�x � �3 � x � 5�2

12 ;

�1, 2, 3, 4��4, �3, �2, �1, 0, 1, 2�;

x � � 23��x � x

� y � �5 � y � 4��x � �3 � x � 3�;range � ��3, �2, 2, 3�Domain � ��3, �2, 0, 2, 5�;

g�x� � 154 x �

134

DW

�2 21 4

2

�2

�4

�4�5 �3 �1�1

�3

�5

345

3 5

y

x

f (x)� x 3 � 1

�2 21 4

�2

�4

�4�5 �3 �1�1

�3

�5

1

345

3 5

y

x

f (x)� 2 � x 2

x

y

f (x) � x 2 � x � 2

x

y

f (x) � x 2

A-30

Answers

5. (a) (b) ; (c) 0; (d)7. (a) 1; (b) all real numbers; (c) 3; (d) all real numbers9. (a) 1; (b) all real numbers; (c) �2, 2; (d)11. (a) �1; (b) ; (c) �4, 0, 3; (d) 13. is a real number and15. All real numbers 17. All real numbers 19. is areal number and 21. All real numbers

23. is a real number and25. is a real number and27. All real numbers 29. All real numbers31. is a real number and 33. All real numbers

35. is a real number and 37. �8; 0; �2

39. 41. 42. 43.

44. 45. R 2; or

46. R or

47.

48. 49. 50.51. is a real number and


1. The graph of looks just like the graph of

but it is moved down6 units.

2. The graph of looks just like the graph of

but it is moved up3 units.

3. The graph of looks justlike the graph of but itis moved up 2 units.

4. 5. �0, � 23��0, 8�

f �x�,g�x�

x

y

g(x) � ax � 2

f (x) � ax

y � �2x,

y � �2x � 3

x

y

y � �2x � 3y � �2x

y � 3x,

y � 3x � 6

x

y

y � 3x y � 3x � 6

� y � y � 0�� y � y � �4�;� y � y � 2�;y � 0�;� y � y

8w2 � w �1481y2 � 180y � 100a2 � 1

14x2 � 57x � 27x4 � x3 � 2x2 � x � 2 �2x � 3x2 � 1

2x � 3;x4 � x3 � 2x2 � x � 2,

w � 1 �2

w � 3w � 1,t � 4

5x � 2

2� y � 2�7� y � 7�

a � 1DW

x � � 54��x � x

x � 52��x � x

x � 1��x � xx � 3

2��x � x

x � 145 �

�x � xx � �3��x � x� y � �2 � y � 2�

�x � �6 � x � 5�� y � y � 0�

� y � �5 � y � 4��x � �4 � x � 3�2 14 ;

�3 10�2 4�1 0

0 �21 �22 03 4

x f �x�

�3 9�2 4�1 1

0 01 12 43 9

x f �x�

�2 �7�1 0

0 11 22 9

x f �x�

anspgs021-030 1/19/06 2:39 PM Page A-30

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N:0-536-47742-6


6. ;

7. ; 8.

9. Slope: �8; y-intercept: 10. Slope: y-intercept:

11. The rate of change is 3 haircuts per hour.12. �0.141 reports per 1000 passengers per year


1. The graph of is the same as the graph ofbut it is moved up 4 units. 2. The graph of

is the same as the graph of but it ismoved down 3 units. 3. The graph of will bethe same as the graph of but it will be moved up8 units. 4. The graph of will be the same as thegraph of but it will be moved down 5 units.


1. The graph of will slant up from left to right. Itwill be steeper than the other graphs. 2. The graph of

will slant up from left to right. It will be less steep than the other graphs. 3. The graph of will slant down from left to right. It will be steeper than the other graphs. 4. The graph of will slant down from left to right. It will be less steep than theother graphs.


1. y-intercept: 3. y-intercept:5. y-intercept: 7.

y-intercept: 9. y-intercept:

11. y-intercept: 13. y-intercept:15. y-intercept: 17.

19. 21. 23. 25.27. or 8% 29. 3.5% 31. The rate of change is5.06 billion messages daily per year. 33. The rate ofchange is �$900 per year. 35. The rate of change is onepoint per $1000 of family income. 37. 39. �1323

40. 41. 42. 2543. Square: 15 yd; triangle: 20 yd

350x � 60y � 12045x � 54

DW

225 ,

m � � 13m � 2

3m � 2m � 13

m � � 12�0, 4

17�m � 0;�0, 12�m � �8;�0, �2�m � 3;

�0, � 83�m � 2

3 ;�0, �9�m � 0.5;�0, �

15�m � �

38 ;�0, �6�

m � �2;�0, 5�m � 4;

y � �0.005x

y � �10xy � 0.005x

y � 10x

y1 � x,y � x � 5

y1 � x,y � x � 8

y1 � x,y3 � x � 3y1 � x,

y2 � x � 4

�0, � 52�

12 ;�0, 23�

m � � 23m � �

13

x

y

(3, 1)(0, 2)

m � �1

x

y

(�1, �1)

(2, �4)

A-31

Chapter 7

44.45.46.

47. R �4; or


1. 2.

3. 4.

5. 6.

7. 8.

x

y

x � �5

x

y

y � 3.6

m � 0

x

y

f (x) � �4

m � 0

x

y

(0, �4)

(�3, 1)

y � �fx � 4

x

y

(0, 5)

(5, 2)

g(x) � �Ex � 5

x

y

(0, �2)

(4, 1)

f (x) � !x � 2

(�4, �5)

x

y

(�2, �2)

(2, 4)

(0, 1)

y � wx � 1

x

y

4y � 12 � �6xy-intercept(0, 3)

x-intercept(2, 0)

a � 10 ��4

a � 1a � 10,

7�2x � 1� �4x2 � 2x � 1��c � d � �c2 � cd � d2� �c � d � �c2 � cd � d2��2 � 5x� �4 � 10x � 25x2�

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42-6


9.

10. (a) Not defined; (b) (c) (d) not defined;(e) (f ) not defined 11. Yes 12. No 13. No14. Yes 15. No


1. 2.

3. 4.

5. 6.

7. 8.




1. D 2. I 3. H 4. C 5. F 6. A 7. G 8. B9. E 10. J

y � (4x � 2)/5

�2 2

�2

2y � 1.2x � 12

�6 16

�20

5


y � 0.4x � 5

�5

�10

20

10


y � (�8x � 9)/3

�3 3

�6

6

y � (5x � 30)/6

�10 10

�10

10y � (�6x � 90)/5

�5

�5

25

25


y � 4.25x � 85

�25�5

5

100


y � �3.2x � 16


�10 10

�20

5

m � 0;m � 0;m � 0;

x

y

8x � 5 � 19

A-32

Answers


1. 3.

5. 7.

9. 11.

13. 15.

17. 19.

x

y

f (x) � �ex � 4

x

y

y � ex � 1

x

y

(0, r)

(Y, 0)5x � 2y � 7

x

y

(0, �3.5)

(2.8, 0)

2.8y � 3.5x � �9.8

x

y

(0, �2)

(3, 0)

2x � 3y � 6

x

y

(0, �3)

(5, 0)

5y � �15 � 3x

x

y

(0, �2)

(�1, 0)

f (x) � �2 � 2x

x

y

(0, 2)

(3, 0)

2x � 3y � 6

x

y

(0, 2)(6, 0)

x � 3y � 6

x

y

(0, �2)

(2, 0)

x � 2 � y

anspgs031-040 1/19/06 2:48 PM Page A-32

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N:0-536-47742-6


21. 23.

25. 27.

29. Not defined 31.

33. 35.

37. 39. Not defined

41. Yes 43. No 45. Yes 47. Yes 49. Yes51. No 53. No 55. Yes 57. 59.60. 61. 62.63. 0.0000213 64. 901,000,000 65. 20,00066. 0.085677 67. 68.69. 70. 71. y � 364�x � 2y � 4�7p�3 � q � 2�

3a�4 � 7b�3�3x � 5y�

9.9902 � 1071.8 � 10�24.7 � 10�55.3 � 1010DW

x

y

7� 3x � 4 � 2x

x

y

2 � f (x) � 5 � 0

m � 0

x

y

y � 0

m � 0

x

y

f (x) � �6

m � 0

x

y

y � �1

m � 0

x

y

x � 1

x

y

5x � 4 � f (x) � 4

x

y

f (x) � ax � 4

x

y

4x � 3y � 12

x

y

x � 2y � 4

A-33

Chapter 7

73. 75. 77. yes79. 81. (a) II; (b) IV; (c) I; (d) III


1. 2. 3.4. 5. 6.7. 8. 9.10. (a) (b)

(c) $210 11. (a) (b) (c) 25.2 in.

Calculator Corner, pp. 534–536

1. 5 2. (a) ; (b) 90%


1. 3. 5.7. 9. 11.13. 15. 17.19. 21. 23.

25. 27. 29.31. 33. 35.37. 39. 41.

43. 45. (a)(b) ; (c) $345

47. (a)(b) ; (c) $425

49. (a)(b) 3.088 trillion prescriptions; 3.85 trillion prescriptions

P�x� � 0.127x � 2.707;

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of months since purchase

Val

ue

100

200

300

400

500

600

700

$800

t

y

V(t) � 750 � 25t

V�t� � 750 � 25t ;

1 2 3 4 5 6 7 8 9 10

Number of hours of moving

Tota

l co

st

100

200

300

400

$500

t

y

C(t) � 40t � 85

C�t� � 40t � 85;y � 52 x � 9

y � 43 x � 6y � 1

2 x � 4y � 13 x � 4

y � 57 x �

177y � �

12 x �

172y � 13x �

154

y � 16 xy � 3

2 xy � 74 x � 7

y � xy � 12 x �

72y � 2

3 x �83

y � �7y � �2x � 16y � x � 6y � �3x � 33y � 5x � 17f �x� � 2

3 x �58

f �x� � � 73 x � 5y � 2.3x � 1y � �8x � 4

y � 2.747311828x � 62.32258065

7 78 ;H�C� � 5

16 C �18 ;

0 1 2 3 4 5 6

40

80

120

160

200

240

280

$320

Number of months of service

To

tal c

ost

t

C(t) � 40t � 50

C (t )C�t� � 40t � 50;y � �2x � 14y � 3x � 7y � �17x � 56

y � � 53 x �

113y � �

23 x �

143y � 8x � 19

y � 3x � 5y � �5x � 18y � 3.4x � 8

m � � 34

y � 0;y � 215 x �

25a � 2

anspgs031-040 1/19/06 2:48 PM Page A-33

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-536

-477

42-6


51. (a) (b) 41.325 sec; 41.1 sec; (c) 2021 53. (a) (b) about 75.8 yr

55. 57. �1 58. 59.

60. 61. $1350


1. True 2. False 3. True 4. True 5. False6. True 7. False 8. False


1. No 2. Yes 3.4. 5. About $6810 6. Yes7. No 8. (a) (b) ; (c) �1; (d) 9. is a real number and10. All real numbers 11. Slope: �3; y-intercept: 12. Slope: y-intercept: 13.14. 15.

16. 17.

18. 19.

20. Perpendicular 21. Parallel 22. Parallel

23. Perpendicular 24.25. 26. 27.28. 29. (a)(b) about 44.37 sec; 44.24 sec 30. The concept ofslope is useful in describing how a line slants. A line withpositive slope slants up from left to right. A line withnegative slope slants down from left to right. The larger theabsolute value of the slope, the steeper the slant.

DWR�x� � �0.064x � 46.8;y � 1

3 x �13

y � � 57 x � 9y � �

32 xy � �3x � 4

f �x� � 4.7x � 23

x

y

f(x) � 4

x

y

x � �3

x

y

f(x) � ex � 3x

y

g(x) � �sx � 4

x

y

(2, 0)

(0, 3)

2y � 6 � 3x

x

y

(4, 0)

(0, 2)

2y � x � 4

m � 113�0, 2��

12 ;

�0, 2�x � 4��x � x� y � 1 � y � 5�

�x � �2 � x � 4�f �2� � 3;f �0� � 7; f ��1� � 12

g�0� � 5; g��1� � 7

4y � 9

x � 32�x � 5�

b � 1DW

M�t� � 0.236 t � 71.8;R�t� � �0.075t � 46.8;

A-34

Answers

31. The notation can be read “f of x” or “f at x” or“the value of f at x.” It represents the output of the functionf for the input x. The notation provides a conciseway to indicate that for the input a, the output of thefunction f is b. 32.


1. [7.1c] 2. [7.1c]

3. [7.1e] (a) 8.666 yr; (b) 1998 4. [7.1a] Yes 5. [7.1a] No6. [7.1b] �4; 2 7. [7.1b] 7; 8 8. [7.1d] Yes9. [7.1d] No 10. [7.1e] (a) About 32 million persons; (b) about 80 million persons 11. [7.2a] (a) 1.2; (b) (c) �3; (d)12. [7.2a] All real numbers 13. [7.2a] is a realnumber and 14. [7.3b] Slope: y-intercept: 15. [7.3b] Slope: y-intercept: 16. [7.3b] 17. [7.3b] 18. [7.3c]

19. [7.4a] 20. [7.4b]

21. [7.4c] 22. [7.4c]

23. [7.4d] Parallel 24. [7.4d] Perpendicular25. [7.5a] 26. [7.5a]

27. [7.5b] 28. [7.5c]

29. [7.5d] 30. [7.5d]

31. [7.5e] (a) (b) 27.57 yr; 28.38 yr32. [7.5d] 33. [7.1b] answers may varyf�x� � 3;24

5

A�x� � 0.115x � 23.2;

y � 3x � 1y � 12 x � 3

y � �32 xy � �4x � 2

f�x� � 5.2x �58y � �3x � 4.8

x

y

2x � �4

x

y

y � f (x) � �3

x

y

f (x) � �sx � 1

x

y

y-intercept(0, 2)

x-intercept(3, 0)

2x � 3y � 6

m �or rate of change� � 45 km�min

m � 0m � 58�0, �

75�

�25;�0, 12��

35;x � �

32�

�x � x� y � �1 � y � 2��x � �3 � x � 4�;

x

y

g(x) � 2 � �x�

x

y

f (x) � �Ex

f �x� � 3.09x � 3.75

f �a� � b

f �x�DW

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ISB

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CHAPTER 8


1. 2. 3. No solution 4. Consistent: 1, 2;inconsistent: 3 5. Infinitely many solutions6. Independent: 1, 2, 3; dependent: 57. (a) �3; (b) , �3; (c) the same, �3

8. , �2

9. (a) , �3; (b) All are �3.

10. , �2


1. 2. 3. 4.


1. consistent; independent 3. consistent;independent 5. consistent; independent7. consistent; independent 9. ; consistent;independent 11. consistent; independent�3, �2�;

�52, �2��2, 1�;

�4, �2�;�1, �2�;�3, 1�;

�3, �1��1, 5��4, �1��2, 3�

x

y

(�2, 0)

f (x) � qx � 1

x

y

(�3, 0)

f (x) � ax � 1

x

y

(�2, 2)f (x) � qx � 3

g(x) � 2

x

y

(�3, �2)

f (x) � x � 1

g(x) � sx

�2, 1��0, 1�

A-35

Chapters 7–8

13. No solution; inconsistent; independent 15. Infinitelymany solutions; consistent; dependent 17.consistent; independent 19. consistent;independent 21. Consistent; independent; F23. Consistent; dependent; B 25. Inconsistent;independent; D 27. 29.30. 31. 33.


1. 2. 3. 4. 5. (a) Nosolution; (b) the same, no solution 6. Length: 160 ft;width: 100 ft


Left to the student


1. 3. 5. 7.

9. 11. 13. 15. No solution17. Length: 40 ft; width: 20 ft 19. 48° and 132°21. Wins: 23; ties: 14 23. 25. 1.3

26. 27. 28. 29.

31. Length: 57.6 in.; width: 20.4 in.


1. 2. 3. 4.5. 6. No solution 7. Infinitely manysolutions 8.9. 10. (a) Length: 160 ft; width: 100 ft; (b) the solutions are the same.


1. We get and Since the graphsare parallel lines, the system of equations has no solution.

2. Each equation is equivalent to or

Since the equations are equivalent, the graphs are the sameand every solution of one is also a solution of the other.Thus we have an infinite number of solutions.


1. 3. 5. 7.

9. 11. 13. 15. Infinitely

many solutions 17. No solution 19.21. 23. 25.27. Length: 110 m; width: 60 m 29. 14° and 76°31. Coach-class: 131; first-class: 21 33. 35. 136. 5 37. 3 38. 291 39. 15 40.41. 53 42. 8.92 43. is a real number and x � �7��x � x

12a2 � 2a � 1

DW

�140, 60��100011 , �

100011 ��

43, �

193 �

�12, �

12�

�11019 , �

1219��4, 6��140

13 , �5013�

�6, 2��12831 , �

1731��1, 3��1, 2�

y � 23 x � 2.y �

2x � 63

,

y � �3x � 2.y � �3x � 5

�2563, 1

7�9x � 4y � 3;9x � 10y � 5,�2, �1�3x � y � 7;2x � 3y � 1,

��138, �118��127, 100��2, 3��1

3, 12��1, 4�

b � 52m � �

12;7

3p �7Aq

�15y � 39

DW

�198 , 1

8��12, 1

2��2, 1��2, �6��2, �2��21

5 , 125 ��2, �3�

�13, 16��3, 2��5, 2��2, 4�

�3, 3�, ��5, 5��2.23, 1.14�y � 38

x �94

y � 35

x �225

DW

�2, �3�;�4, �5�;

anspgs031-040 1/19/06 2:48 PM Page A-35

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N:0

-536

-477

42-6


44. Domain: all real numbers; range: 45. 46.47. 49.51.


1. White: 22; red: 8

2. Kenyan: 8 lb; Sumatran: 12 lb3. $1800 at 7%; $1900 at 9%

4. Attack: 15 qt; Blast: 45 qt

5. 280 km

6. 180 mph

q � �13p � 2,

B � 4A � 2,�23.12, �12.04�y � �

72

x � 36y � � 35

x � 7� y � y � 5�

A-36

Answers


1. G 2. E 3. D 4. A 5. J 6. B 7. C 8. I9. F 10. H


1. 32 brushes at $8.50; 13 brushes at $9.753. Humulin: 21 vials; Novolin Velosulin: 29 vials5. 30-sec: 4; 60-sec: 8 7. 5 lb of each 9. 25% acid: 4 L;50% acid: 6 L 11. 10 silk neckties 13. $7500 at 6%; $4500 at 9% 15. Whole milk: cream: 17. $5 bills: 7; $1 bills: 15 19. $7400 at 5.5%; $10,600 at 4% 21. 375 mi 23. 25. 144 mi27. 2 hr 29. 31. About 1489 mi 33.35. 36. 37. 38. 33 39.40. 41. 42. 0.2 43. 44.45. 46. 3993 47. 49. City: 261 mi; highway: 204 mi 51. Brown: 0.8 gal; neutral: 0.2 gal


1. 2. 3.


1. 3. 5. 7.9. 11. 13. 15.17. 19. 21.

23. 25. 27. 28.

or 29. or 30.

or 31. 32. 33. Slope:

y-intercept: 34. Slope: �4; y-intercept:

35. Slope: y-intercept: 36. Slope: 1.09375; y-intercept: 37.


1. 64°, 32°, 84° 2. $4000 at 5%; $5000 at 6%; $16,000 at 7%


1. 20-oz: 11; 32-oz: 15; 40-oz: 8 3. 32°, 96°, 52°5. �7, 20, 42 7. Automatic transmission: $865; powerdoor locks: $520; air conditioning: $375 9. A: 1500; B: 1900; C: 2300 11. First fund: $45,000; second fund:$10,000; third fund: $25,000 13. Asian–American: 385;African–American: 200; Caucasian: 154 15. Roast beef: 2;baked potato: 1; broccoli: 2 17. Par-3: 6; par-4: 8; par-5: 419. Two-pointers: 32; three-pointers: 5; foul shots: 1321. 23. 25. At most 26. At least27. Linear 28. Negative 29. Consistent30. Perpendicular 31. y-intercept 32. Horizontal33. 180° 35. 464

DWDW

�1, �2, 4, �1��0, �3.125��0, �2�2

5;

�0, 5��0, �54��

23;

y �c � Ax

By �

Ax � cB

c �2Ft

d �tc � 2F

t,

2Ft

� dc �2F � td

t,

Q4

� b

a �Q � 4b

4,a �

F3b

DW�4, 1, �2�

�15, 33, 9��12, 2

3, �56��1

2, 13, 1

6��1

2, 4, �6��2, 2, 4��3, 0, 4��2, 4, 1��3, �4, 2��3, 1, 2��2, 0, 1��1, 2, �1�

�20, 30, 50��2, �2, 12��2, 1, �1�

4 47 L�12h � 7

�17�4�238a � 7�15�3�11�7

DW1 13 hr

14 km�h

30 1013 lb169 3

13 lb;w r 30

$18.95 $19.50

18.95w 19.50r 572.90

WHITE RED TOTAL

w � r � 30

� 572.9018.95w � 19.50r

x y $3700

7% 9%

1 yr 1 yr

0.07x 0.09y $297

FIRSTINVESTMENT

SECONDINVESTMENT TOTAL

x � y � 3700

� 2970.07x � 0.09y

a b 60

2% 6% 5%

0.02a 0.06bor 3

0.05 � 60,

ATTACK BLAST MIXTURE

a � b � 60

0.02a � 0.06b � 3

d 35 km�h t

d 40 km�h t � 1

DISTANCE RATE TIME

d � 35t

d � 40�t � 1�

d 4 hr

d 5 hrr � 20

r � 20

DISTANCE RATE TIME

d � 4�r � 20�

d � 5�r � 20�

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1. (a) (b)(c) (d) a loss of $16,000; a profit of$176,000; (e) , break-even point:

2.


1. (a) (b)3. (a) (b)5. (a) (b)7. (a) (b)9. (a) (b)11. (a) (b)(c) (d) $112,500 profit; $4500 loss; (e) 13. (a)(b) (c) (d) $19,596 profit;$4404 loss; (e) 15.17. 19. 21.23. 25. Slope: y-intercept: 26. Slope:

y-intercept: 27. Slope: 1.7; y-intercept:

28. Slope: y-intercept:


1. True 2. False 3. False 4. True 5. True


1. consistent; independent 2. Infinitely manysolutions; consistent; dependent 3. No solution; inconsistent; independent 4. 5. No solution

6. 7. 8. 9.10. Infinitely many solutions 11. CD: $14; cassette: $912. 5 L of each 13. 14.

15. 16. 17.

18. 19. $20 bills: 5; $5 bills: 15; $1 bills: 1920. (a) (b)(c) (d) $115,000 profit; $10,000 loss; (e) 21. 22. The comparison is summarized in the table in Section 8.3. 23. Manyproblems that deal with more than one unknown quantityare easier to translate to a system of equations than to asingle equation. Problems involving complementary orsupplementary angles, the dimensions of a geometricfigure, mixtures, and the measures of the angles of a triangleare examples.

DW

DW�$3, 81��280, $84,000�P�x� � 125x � 35,000;

R�x� � 300x;C�x� � 35,000 � 175x;90�, 67 1

2 �, 22 12 �

�2, 13, �

23��2, 0, 4��

73, 125

27 , 2027�

�10, 4, �8�5 12 hr

�2, 2��7617, �

2119��37

19, 5319��

1115, �

4330�

�25, �

45�

��2, 1�;

�0, 4��43;

�0, 49��0, 137 �

�67;�0, 8

5�35 ;DW

�$10, 1070��$50, 6250��$22, 474��$70, 300��1367, $24,606�

P�x� � 12x � 16,404;R�x� � 18x;C�x� � 16,404 � 6x;�500, $42,500�

P�x� � 45x � 22,500;R�x� � 85x;C�x� � 22,500 � 40x;�2600, $325,000�P�x� � 75x � 195,000;

�889, $35,560�P�x� � 18x � 16,000;�125, $12,500�P�x� � 80x � 10,000;

�2400, $144,000�P�x� � 50x � 120,000;�6000, $420,000�P�x� � 45x � 270,000;

�$14, 356�

�5000, $180,000�

Units sold

100,000

0

�100,000

200,000

300,000

$400,000

10,000 20,000

P(x)

C(x)R(x)

x

y

P�x� � 16x � 80,000;R�x� � 36x;C�x� � 80,000 � 20x;

A-37

Chapters 8–9

24.

or

dimes: 13; quarters: 725. and 26. (a) 165.91 cm; (b) 160.60 cm; (c)(d) 120.857 cm


1. [8.1a] consistent; independent 2. [8.1a] Nosolution; inconsistent; independent 3. [8.1a] Infinitely many solutions; consistent; dependent 4. [8.2a]

5. [8.2a] 6. [8.3a] 7. [8.3a] No

solution 8. [8.4a] 34% solution: 61% solution: 9. [8.4a] Buckets: 17; dinners: 11 10. [8.4b]

11. [8.2b], [8.3b] Length: 93 ft; width: 51 ft12. [8.5a] 13. [8.7b] 14. [8.6a] 3.5 hr 15. [8.7a] (a) (b)(c) (d) $20,000 profit; $30,000 loss; (e) 16. [8.2a], [8.3a]

CHAPTER 9


1. Yes 2. No 3. Yes 4. 5.6. 7. 8. ��30, 30��10, ��2, 6�

��, �2��4, 5�

m � 7; b � 10�800, $64,000�P�x� � 50x � 40,000;

R�x� � 80x;C�x� � 40,000 � 30x;�$3, 55��2, �

12, �1�

120 km�h71 1

9 mL48 8

9 mL;

��32, �

32��15

7 , �187 �

�3, �113 �

��2, 1�;

�120.857, 308.521�;�1, 3��0, 2�

d � q � 6;d � q � 20

25d � 10q � 90 � 10d � 25q; d � q � 20,

9. , or ;

0 3

�3, ��x � x � 3� 10. , or ;

0 3

��, 3��x � x � 3�

11. , or ;

0�2

��, �2��x � x � �2� 12. , or ;

0

Í

��, 310�� y y �

310�

15. , or 16. , or

17. , or 18. 19.20. 21.22. 23. 24.25. 26. 27.28. , or years after 200929. For , plan A is better.


1. F 2. I 3. C 4. E 5. D 6. J 7. O 8. M9. B 10. L


1. No, no, no, yes 3. No, yes, yes, no, no 5.7. 9. 11. 13. ��2, ��2, 5��8, �4��3, 3�

��, 5�

�n � n 100 hr��t � t � 9.1�

p 37d � 11.6%c � $135,000n � �8w 110 lbd � 25 mi

50 min t 60 minp � $21,000d $4000s 88�17

9 , �� y � y 179 �

��, 12��x � x

12��, �

15�� y � y � �

15�

13. , or

0�5

��, �5�� y � y �5� 14. , or

0 4 12

�12, ��x � x 12�

anspgs031-040 1/19/06 2:49 PM Page A-37

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42-6


A-38

Answers

9.

10. ;

11. , or ;

12. , or ;

13. , or ��, �5� � ��2, ��x � x � �2 or x �5�0�2

r

��, �2� � �72 , ��x � x

72 or x �2�

0 1 6

��, 1� � �6, ��x � x 1 or x 6�

��, �2� � �4, ��0�2 4

A

B

15. , or

0�1

��1, ��x � x � �1� 17. , or

60

��, 6�� y � y 6�

19. , or

21. , or

0�4

��4, ��t � t �4�0�10

�22

��, �22��a � a � �22�

23. , or

�6 0

��6, �� y � y � �6� 25. , or

0 4

9

��, 9��x � x � 9�

27. , or

0 3

�3, ��x � x 3�

29. , or

�60 �30 0

��, �60��x � x �60� 31.

30

�x � x � 3�, or �3, ��

33. , or

35. , or 37. , or

39. , or 41. , or 43. , or

45. , or 47. , or

49. , or 51. , or

53. , or 55. , or

57. , or 59. , or

61. , or 63. , or

65. , or 67. , or 69. , or 71. 73.75. 77.79. 81. 83.85. (a) 8.398 gal, 12.063 gal, 15.728 gal; (b) years after 199987. 89. 90.91. 92.93.94. All real numbers 95. All real numbers96.97. (a) ; (b) 99. True101. All real numbers 103. All real numbers


1. 2.

3. ;

4. , or ;

5. ;

0�3

�x � x �3�

�5 10

0 4

��5, 10��x � �5 x � 10�

��1, 4��1 40

A

B

�0, 3�

� p � p 10�� p � p � 10��x � x is a real number and x � 2

3�

�x � x is a real number and x � �8�t 2 � 7st � 18s26a2 � 7a � 55

6r 2 � 23rs � 4s23x2 � 20x � 32DW

�s � s � 8�� p � p � 80��n � n � 25��S � S � $7000��B � B $11,500�

�S � S 84��W � W �approximately� 189.1 lb��, 2��a � a � 2�

�� 5131 , ��x � x �

5131��, 11

18��x � x 1118�

�132 , ��x � x

132 ��, 8��x � x 8�

��, 47��x � x �

47��, 5�� y � y 5�

�2, ��x � x 2��, �3��r � r �3��7

3 , ��m � m �73��

217 , ��x � x � �

217�

��, � 752 �� y � y � �

752 ��, 1

2��x � x �12�

�11.25, ��x � x 11.25��2

3 , �� y � y �23��, �3�� y � y � �3�

��, 6��x � x 6��, 56��x � x �

56�

��, 0.9��x � x � 0.9�

6. �

14. All real numbers;

0

15. �s � 8 s 19�


1. 3. 5.7. 9. 11.13. ;

15. ;

17. , or ;

�4 50

��4, 5��x � �4 � x 5�

�1, 6�60 1

��4, 1��4 10

�3, 5, 7��a, b, c, d, f, g��9, 10, 11, 13��b��9, 11�

19. , or ;

0 2

�2, ��x � x 2� 21. �

23. , or 25. , or 27. , or

29. , or

31. , or

33. , or

35. , or

37. , or

39. ;

41. ;

43. , or ;

45. , or ;

47. , or ;

49. , or 51. All real numbers, or 53. , or

55. , or

57. , or

59. 61. Between 23 and 27 beats�d � 0 ft � d � 198 ft��, �

132 � � �29

2 , ��x � x � � 132 or x

292 �

��, 794 � � �89

4 , ��x � x 794 or x �

894 �

��, �4� � �2, ��x � x �4 or x � 2��, ��

��, � 54� � ��

12 , ��x � x � �

54 or x � �

12�

0�3

��3, ��x � x �3�0 4

e

��, 52� � �4, ��x � x �

52 or x 4�

0�5 �1

��, �5� � ��1, ��x � x �5 or x � �1�

��, �3� � �1, ��0�3 1

��, �2� � �1, ��0�2 1

�� 72 , 37

2 ��x � 72 x

372 �

�10, 14��x � 10 x � 14��

32 , 9

2��x � 32 � x

92�

��1, 6��x � �1 x � 6��

53 , 4

3��x � 53 � x �

43�

��1, 5�� y � �1 y � 5��6, 2��x � �6 x � 2��8, 6��x � �8 x 6�

7. , or 8. �0, 1, 3, 4, 7, 9��2, 6��x � 2 � x � 6�

anspgs031-040 1/19/06 2:49 PM Page A-38

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63.65. 67. 69.70. 71. 72.73. 74. 75.76. 77.78. 79. , or

81. , or 83. , or

85. , or 87. True 89. False91. All real numbers;


1. 2. 3. 4. 5. 6. 29

7. 5 8. 9. ;

�6 60

�6, �6�� p�

9�x�7�a�b25a2�b�x87�x�

��10, 18��x � 10 x � 18�

�� 18 , 1

2��x � 18 x

12��2

5 , 45��x 2

5 � x �45�

��4, 1��x � �4 x � 1�13x2 � 37xy � 6y221x2 � 59x � 4025y2 � 35y � 6

6a2 � 7ab � 5b2y � �x � 2y � �4x � 7y � �11x � 29�4, �4��3, �5�

��1, 2�DW�d � 250 mg d 500 mg��W � 139.9 lb W 189.1 lb�

A-39

Chapter 9

89. , or

91. , or

93. , or

95. , or 97. , or 99. , or 101. , or

103. , or 105. , or

107. 109. Union 110. Disjoint 111. At least112. 113. Absolute value 114. Equation115. Equivalent 116. Inequality117. 119. , or

121. 123. 125. All real numbers

127. 129. 131.


1. No 2. Yes 3.

4. 5.

6. 7.

8. 9.

x

y

(0, 0) (4, 0)

(0, 3)(P, 3)

(4, f)

x

y

x

y

x

y

y > �4

x

y

x , 3

x

y

4x � 3y ˘ 12

x

y

6x � 3y , 18

�x � 3� � 5�x� 6�x� 3

��1, � 14�

��5, ��x � x �5��d 5 12 ft � d � 6

12 ft�

�a, b�DW

�0.49705, 0.50295��x � 0.49705 � x � 0.50295��1

2 , 52��x 1

2 � x �52�

��12, 2��m � �12 � m � 2��, �

215� � �14

15 , ��x � x � � 2

15 or x 1415�

��5, 19��x � �5 x 19��, �

256 � � �23

6 , ��x � x � � 256 or x

236 ��

92 , 6��x �

92 x 6�

��, � 54� � �23

4 , ��x � x � � 54 or x

234 �

��, � 43� � �4, �� y � y �

43 or y � 4�

10. �

11. 12. 13. 14.15. 16. 17. 18.19. 20. ;

21. , or ;

22. , or ;

23. , or ;

24. , or

25. , or ;


1. 3. 5. 7. 9. 11.

13. 15. 17. 38 19. 19 21. 6.3 23. 5

25. 27. 29. 31.33. 35. 37.39. 41. 43. 45.47. 49. 51. 53.

55. 57. 59. 61.

63. All real numbers 65. 67. 69.71. , or 73. , or 75. , or 77. , or

79. , or

81. , or

83. , or 85. , or

87. , or �� 72 , 1

2��x � 72 � x �

12�

��9, 15�� y � �9 y 15��, �

54� � �23

4 , ��x � x � � 54 or x

234 �

��, � 32� � �17

2 , �� y � y � 32 or y �

172 ��

12 , 7

2��x � 12 � x �

72�

��6, �2��x � �6 � x � �2��0, 2��x � 0 x 2�

��, �2� � �2, ��x � x � �2 or x 2��3, 3��x � �3 x 3�

�32, 83��24

23 , 0�� 32�

�5, � 35��3

2��34 , �

112 ��

1354 , �

754�

��72 , �

52��2, �12��8, �7�

�2, �2��7, �7��8, �8��291, �291��11, �11��23

4 , � 54��7

2 , � 12�

�15, �9��0��3, �3�

y2

34�x�

x2

� y�2�x�

6� y�2x22x29�x�

0 1�g

��, � 73� � �1, ��x � x � �

73 or x 1�

�1, 113 ��x � 1 � x �

113 �0�2 5

��2, 5��x � �2 x 5�0�5 5

��, �5� � �5, ��x � x � �5 or x 5�0�5 5

��5, 5��x � �5 x 5�0�5 5

�5, �5�� 72�

�74 , �

16��

133 , 7��3, 5�

�4, �4��174 , �

174 ��2, �2��0�

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42-6


10.




1. D 2. B 3. E 4. C 5. I 6. G 7. F 8. H9. A 10. J


1. Yes 3. Yes 5.

7. 9.

11. 13.

x

y

3x � 4y < 12

x

y

x � y , 4

x

y

y . x � 2

x

y

y , x � 1

x

y

y . 2x

(2, 1)(0, 2)

(0, 0) (3, 0) x

y

A-40

Answers

15. 17.

19. 21.

23. 25. F27. B29. C

31. 33.

35. 37.

x

y

(3, �7)

2

8

(1, �2)

x

y

x

y

(q, q)

x

y

(1, 1)

x

y

2x � 3y < 6

x

y

y . 2

x , 5

x

y

x

y

3x � 2 < 5x � y

x

y

2y � 3x . 6

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39. 41.

43. 45.

47. 49. 50.

51. 52. 53.54. 55. 2 56. 3 57. 1 58. 8 59. 460. or 61. 6 62. 0.263.

or

or

65. Left to the student 67. Left to the student69. Left to the student


1. False 2. True 3. True 4. True 5. False


1. 2.3. ;

4. ;

5. , or 6. , or 7. , or 8. , or 9. , or

10. , or 11. , or

12. , or

13. , or 14.15. $10,000 16. 17.18. ;

19. ; 20.

21. , or ��7, 2��x � �7 � x � 2�

��, �2� � �5, ��0�2 5

��2, 5��2 50

�1, 2, 3, 5, 6, 9��1, 5, 9��t � t � 4

14 hr��, �

52��x � x � �

52�

��10, �� y � y � �10��, �3��x � x � �3��, �

65�� y � y � �

65�

��3, ��x � x � �3��30, �� y � y � �30��4, �� y � y � �4��7, �� y � y � �7�

��, �21��a � a � �21�

�1, ��0 1

��, �2�0�2

��, 40��8, 9�

20

40

�20

�40

�20 20�40

w

hw � h � 502w � 2h � 30 � 130,w � h � 32,w � h � 30 � 62,h � 0,w � 0,

2�1 � a��2 � 2a�,y � 6

y � � 32

x �52y � 9

2 x � 9y � �3x � 15

y � 38

xy � 12

x �72

DW

x

y

(5, 0)(0, 0)

(0, 4)

6

8

(Ç, Ç)40 2411 11

x

y

(0, 6) (4, 4)

(6, 0)(0, 0)

8

8

x

y

(w, �q)

x

y

(0, 1)

(2, �3)

(2, 5)

A-41

Chapter 9

22. , or 23. , or 24. , or

25. , or 26.

27. 28. 29. 62 30. 6, �6 31.

32. 33. 34. , or

35. , or

36. , or 37.38. , or approximately between 1991 and 1996

39. 40.

41. 42.

43. 44.

45. (1) , not . (2) Thiswould be correct if (1) were correct except that theinequality symbol should not have been reversed. (3) If (2) were correct, the right-hand side would be �5, not8. (4) The inequality symbol should be reversed. Thecorrect solution is

.

46. “Solve” can mean to find all the replacements thatmake an equation or inequality true. It can also mean toexpress a formula as an equivalent equation with a givenvariable alone on one side.47. , or ��

83 , �2��x ��

83 � x � �2�

DW

x �254

�4x � �25 7 � 3x � x � 18

7 � 9x � 6x � �9x � 18 � 10x 7 � 9x � 6x � �9�x � 2� � 10x

�9x � 2�9�x � 2� � �9x � 18DW

�2 21 4

�2�1

�4

�4�5 �3 �1

�3

�5

1

345

3 5

y

x

(5, 2)

(1, �2)

(�3, 2)

x

y

(2, �3)

x

y

x

y

x � y > 1

x

y

y < 0x

y

2x � 3y , 12

�t � 3.23 � t � 7.85��, �

113 � � �19

3 , ��x � x � � 113 or x �

193 �

��, �3.5� � �3.5, ��x � x � �3.5 or x � 3.5��

172 , 7

2��x �� 172 � x �

72��14, 4

3��9, �5�

4� y�

2�x�y2

3�x�

��, �6� � �8, ��x � x � �6 or x � 8�

��, �11� � ��6, ��x � x � �11 or x � �6��, �3� � �1, ��x � x � �3 or x � 1�

�� 54 , 5

2��x �� 54 � x �

52�

anspgs041-050 1/19/06 2:52 PM Page A-41

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-477

42-6



1. [9.1b] 2. [9.1b] 3. [9.1c] ;

4. [9.1c] ;

5. [9.1c] 6. [9.1c] , or

7. [9.1c] , or 8. [9.1c] , or

9. [9.1c] , or

10. [9.1c] , or 11. [9.1d]

12. [9.1d] 13. [9.2a] ;

14. [9.2b] ;

15. [9.2a] , or 16. [9.2a] , or

17. [9.2a] , or

18. [9.2b] , or

19. [9.2b] All real numbers, or 20. [9.2b] , or

21. [9.3a] 22. [9.3a] 23. [9.3b] 8.4

24. [9.2a] 25. [9.2b] 26. [9.3c] 27. [9.3c] 28. [9.3d]

29. [9.3c] 30. [9.3e] 31. [9.3e] 32. [9.3e]

33. [9.3e] 34. [9.4b]

35. [9.4c] 36. [9.4c]

37. [9.3e] 38. [9.2a]


1. [7.5e] (a) (b) 3.50 min; 3.49 min2. [7.3c] 4% per year 3. [1.8b] 47b � 51

R�x� � �0.006x � 3.85;

�x � 15 � x �

45�, or �1

5 ,

45��

(�2, 0)

(��, ��)9 4

1 2

�2 21 4

�2�1

�4

�4�5 �3 �1

�3

�5

1

32

45

5

y

x

(4, �1)

x

y

x

y

x � 6y , �6

�x � x � � 135 or x �

75�, or ��, �

135 � � �

75

, ��x � �99 � x � 111�, or ��99, 111��x � x � �3 or x � 3�, or ��, �3� � �3, ��

�x �� 78 � x �

118 �, or ��

78

,

118 ��

�1��6, 12��9, 9��1, 3, 5, 7, 9, 11, 13��3, 5�

2�x�7�x�

��, 3� � �6, ��x � x � 3 or x � 6��, ��

��, �4� � �� 52 , ��x � x � �4 or x � �

52�

�� 25 , 9

5��x �� 25 � x �

95�

��1, 6��x � �1 � x � 6��4, ��x � x � 4�

��, �3� � �4, ��0�3 4

��3, 4��3 40

�d � 33 ft � d � 231 ft��h � h � 2

110 hr��, 7

4��x � x �74�

�52 , ��x � x �

52�

�1, �� y � y � 1��, 11

5 ��a � a �115 �

��50, �� y � y � �50��x � x � 10�, or �10, ��

��, �2�0�2

��, 6�60

��4, ��3, 2�

A-42

Answers

4. [1.8d] 224 5. [4.2a, b] 6. [4.4c]

7. [4.6c]

8. [4.6a] 9. [6.1d]

10. [6.2b] 11. [6.5a]

12. [6.5b] 13. [6.6a]

14. [6.6a] 15. [4.8b]

16. [2.3c] 17. [2.4b]

18. [9.2a] or

19. [9.1c] or

20. [9.3e] or 21. [9.3c] 22. [9.3e] or or 23. [8.2a], [8.3a] Infinite number of solutions24. [8.5a] 25. [5.8b] 26. [5.8b] 27. [6.7a] No solution 28. [6.7a]

29. [2.4b] 30. [8.2a], [8.3a]

31. [8.5a] 32. [5.1b] 33. [5.1c] 34. [5.2a] 35. [5.3a], [5.4a] 36. [5.5d] 37. [5.5b] 38. [5.6a] 39. [5.6a] 40. [5.7a] 41. [5.3a], [5.4a] 42. [3.2a] 43. [3.3b]

44. [9.4b] 45. [9.4c]

46. [7.5b] 47. [7.5d] 48. [2.5a] 22.5% 49. [8.6a] Win: 38; lose: 30; tie: 1350. [6.9b] About 99 lb 51. [7.2a] is a real numberand and or 52. [7.2a] Domain: range: 53. [6.7a] (a)54. [6.5a] (d) 55. [5.8b] (c) 56. [6.8a] (b)57. [8.6a] 58. [6.7a] All realnumbers except 9 and 59. [5.8b] 0, 14 , � 1

4�5a � 1, b � �5, c � 6

��2, 4��5, 5�;��, �5� � ��5, 5� � �5, ��x � 5�,x � �5

�x � x

y � 12 x �

52y � �

12 x � 1

(e, w)

x

y

x

y

x � 3y , 4

x

y

3x � 18 � 0

x

y

y � �5x � 4

�4x � 1� �5x � 3�x2�x2 � 1� �x � 1� �x � 1��0.3b � 0.2c� �0.09b2 � 0.06bc � 0.04c2�8�2x � 1� �4x2 � 2x � 1�

�t � 8�2�4y � 9� �4y � 9��2x � 5� �3x � 2�

�x � 6� �x � 14��2a � 1� �4a2 � 3�2x2�2x � 9��5

8 , 116 , �

34�

��2, 1�a �bP

3 � P

�1�2, 72

14�3, 2, �1�

��, �2.1� � �2.1, ��x � 2.1�,�x � x � �2.1

43 , 83��2, 5��x � �2 � x � 5�,

��79 , ��x � x � �

79�,

��3, �32��x � �3 � x � �

32�,C � 5

9 �F � 32�152

2x2 � 11x � 23 ��49

x � 2y3 � 2yy3 � 1

y � 2x3y � x

4x � 1�x � 2� �x � 2�

6x � 1320�x � 3�

3x � 5x � 4

y � 23

15a2 � 14ab � 8b2

36m2 � 12mn � n2

16p2 � 8px6

4y8

anspgs041-050 1/19/06 2:52 PM Page A-42

ISB

N:0-536-47742-6


CHAPTER 10


1. 2. 3. 4. 1 5. 6 6.7. 0.08 8. (a) 4; (b) (c) does not exist as a realnumber 9. (a) 7; (b) (c) does not exist as a realnumber 10. (a) 12; (b) (c) does not exist as a real number 11. 12. 13. 1.2 14. 4.12315. 6.325 16. 33.734 17. 18. 0.793

19. 20. 21. 22. 2;

does not exist as a real number23. does not exist as a real number24. Domain

25. Domain

26. 27.

28. 29. 24 30. 31. 32.33. 34. 35. 36.37. 38. 39.40. 41. 3 42.43. 44. 45. 0 46. 47. 48. 349. 50. Does not exist as a real number 51. 052. 53. 54. 55. 56.


1. 6.557 2. 102.308 3. 4.5. 7.469 6. 9.283 7. 2.012 8. 9. 0.77510. 0.775 11. 3.162 12. 4.378


1. 3. 5. 7. 9. 1411. 0.06 13. Does not exist as a real number

15. 18.628 17. 1.962 19. 21.

23. 0; does not exist as a real number; doesnot exist as a real number25. does not exist as a real number;

12 27. Domain29. About 24.5 mph; about 54.8 mph31. 33.

x

y

F(x) � �3œx

x

y

f (x) � 2œx

� �x � x � 2� � �2, ��11 3.317;11 3.317;

20 4.472;

xy � 1

y2 � 16

�7620, �2012, �124, �4

�2.812�0.804�96.985

3xx � 3�x � 3��x�2�x � 2��3

3x � 2�2xyx�33 7 1.913�3; 0; 3

�5 �1.710;

�742�x � 2�3y

�4�x � 3�7�y � 5�2�x � 2��x � 7�4�y�5�y��y�

x

y

f (x) � 2œx � 3

x

y

g(x) � �œx

� �x � x � �32� � ��

32 , ��

� �x � x � 5� � �5, ��7 �2.646;�2;

22 4.690;

yy � 3

28 � x�5.569

�29.455�0.95

8

�12;�7;

�4;

91011, �116, �63, �3

A-43

Chapters 9–10

35. 37.

39. 41.

43. 45. 47. 49. 51. 353. 55. 57. 59. 2; 3; 61. 63.65. 67. 69. 71. 73. 675. 77. 79. 81. 83.84. 85. 86. 4, 9 87. 88.89. 0, 90. 0, 1 91. 92.93. Domain 95. 1.7; 2.2; 3.297. (a) Domain: range: (b) domain:

range: (c) domain: range:(d) domain: range: (e) domain:

range:


1. 2. 3. 2 4. 5 5.

6. 7. 8. 9.

10. 11. 4 12. 32 13. 14.

15. 16. 17. 18. 19.

20. 21. 22. 23. 24.

25. 26. 27. 28. 29.30. 31. 32. 33.34. 35. 36.


1. 3.344 2. 3.281 3. 0.283 4. 11.0535. 6. 2


1. 3. 2 5. 7. 8 9. 343 11.13. 15. 17. 19.

21. 23. 25. 27. 29.2a3/4c2/3

b1/21

�2rs�3/43

x1/41

100013

�8x2y�5/7�3mn�3/2�xy2z�1/5181/3171/25 a3b37 y

5.527 � 10�5

10xa20b12c46 m

7 5m4 ab6 x4y3z54 63ab2a2b4 xy22x

aq1/8

p1/392/551/3714/15

� 7n11m�2/37p3/4

q6/51

27

1�3xy�7/8

12

67/5�7abc�4/35 x3

7�2ab�1/4�x2y16�1/5

19�ab�1/3�19ab�1/3

5 a3b2c3a4 y

�0, ��3, ��;�0, ��;�0, ��;��, 2�;

��3, ��;��, ��;��, ��;��, ��;��, ��;

� �x � �3 � x � 2� � ��3, 2�10a10b9a9b6c155

2

52�2, 53�

72 , 72�1, 0

�2, 1DWx � 2y�a � b�5�a��x��

23�1

�5�4; �10�1; �3�20, or 3 20 2.7144;

�4�2;0.7�x � 1��6�4x�x � 2��p � 3�12�c�4�x�

x

y

g(x) � œ3x � 9

x

y

f (x) � œ12 � 3x

x

y

f (x) � œx � 2

x

y

f (x) � œx

anspgs041-050 1/19/06 2:52 PM Page A-43

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-536

-477

42-6


31. 33. 35. 37.

39. 41. 43. 45.

47. 49. 51. 53.

55. 57. 59. 61. 63.

65. 67. 69. 71.73. 75. 77. 79.

81. 83. 85. 87.

88. 89. 90.91. Left to the student


1. 2. 3. 4.

5. 6. 7. 8.9. 10. 11. 12.13. 14. 15. 16.17. 18. 19. 5 20. 21.

22. 23. 24. 25. 26.


1. 3. 5. 7.9. 11. 13. 15.17. 19. 21. 23.25. 2 27. 29. 31.33. 35. 37. 39.41. 43. 45. 47.49. 51. 53. 55. 3 57.

59. 61. 63. 65.

67. 69. 71. 73. 75. 77.

79. 81. 83. 85.

87. 89. 91. �10, 9 92. Height: 4 in.;

base: 6 in. 93. 8 94. 95. No solution96. No solution 97. (a) 1.62 sec; (b) 1.99 sec; (c) 2.20 sec99.


1. 2. 3.4. 5. 6.7. 8.9. 10. 11.12. 13. 58 � 12620 � 4y5 � y2

p � q�33ab � 43a � 63b � 24�4 � 96a3 3 � 3 2a2

56 � 3142x � 1�3y � 4�3 y2 � 2y2195104 5x � 7132

2yz2z

152

DWx26 x

yz2

2x5 x3

y23x2

3a3 a2b

5yyx2

7y

53

47

56

12x2y5

12a5

1

6 a2x2y24xy23 a2b

y7y532xy6 xy5

b10b9a4 a4 126 200

4a3b6ab44 42y33 2a3 105bc22b3x42303

310522xy35 3x23x2y24 3y2

4a2b24 52t 23 10t 23x23 2x2

6x2553 231026

12xy2

3x3 2x2

5x

1056

207

5a56xy73ab3x3 4y6y7323xy23 3xy223 2

2bc3ab2a3b6y10323 10426 25x3y26 500

3 10pq

4 282121pq133

�� 118

, 38��

152

, 52��40, 40�

�� 47

, 2�DW30 d35

c9912x4y3z2

18ma6b126 4x5

2055 74

1274 532c2d3x2y33 2x

2a5b51x3x53 a

1x2/7a8/3b5/2a23/1263/28

4.91/271/457/85ac1/2

3

7xz1/3

x4

21/3y2/7x2/3�8yz7x �3/5

A-44

Answers


1. 3. 5. 7. 9.11. 13. 15. 17.19. 21. 23.25. 27. 29.31. 33. 35.37. 39. 41.43. 45. 47. 49. 151. 53. 44 55. 1 57. 3 59.61. 63. 65. 67.69. 71.73. 75. 77.

79. 80. 81.

82. 83. 84.

85. 86. 87. 5

88. or

89. or

90. �12, 91. Domain 93. 6

95. 97.


1. 2. 3. 4. 5.

6. 7. 8. 9.

10. 11. 12.


1. 3. 5. 7. 9.

11. 13. 15. 17.

19. 21. 23.

25. 27.

29. 31. 33. 35. 30

36. 37. 1 38. 39. Left to the student

41.


1. 100 2. No solution 3. 1 4. 2, 5 5. 4 6. 177. 9 8. 27 9. 5 10. About 11. About 88°F

1149.9 ft�sec

�3a2 � 3

a2 � 3

x � 2x � 3

�195

DWx � 2xy � y

x � y36 � 4

2

6 � 5a � a9 � a

15 � 20 � 62 � 830�77

�235 � 22154 � 910

264 2xy2x2y

3 100xy5x2y

15x10

4 s3t 3

sty3 9yx2

3x2

3 75ac2

5c23 6

3215

3522

215

3

�7 � 626 � 222 � 3

2 � 3

3 � y

3 � ya � bc2 � b

73 2x2y2y2

x3 12x2y2

2y4 56

223ab

3b3 10

210

5

33 3 � 23 9 � 814 � 215 � 62 � 230

� ��, �� 25

��, � 293 � � �5, ��x � x � �

293 or x � 5�,

�� 293

, 5��x � � 293 � x � 5�,

� 293

,a2b2

b2 � ab � a2pq

q � p

xx � 1

4�3x � 1�3�4x � 1�

� y � 3� � y � 3�y � 3

a � 2�a � 2� �a � 4�

ax�x2 � 4�

�x � 4� �x � 3�

DW5 72 � 3 � 5 24 � 5 817 � 4323 9 � 33 6 � 23 4a � 3a � 2a � 6

�67 � 331 � 5a � b�19�12

3a3 2�6215 � 63�12 � 636 � 2145 � 10�x � 3�x � 13a � 1�x � 1�3 6x

153 42 � 32�21x � 1�3x�2 � x�3 3x�1 � 6a�5a292

93 2122238521363 3�86133 y3 7115

anspgs041-050 1/19/06 2:53 PM Page A-44

ISB

N:0-536-47742-6





1. 3. 5. 57 7. 9. 11. No solution13. 3 15. 19 17. 19. 21. 9 23. 1525. 2, 5 27. 6 29. 5 31. 9 33. 7 35.37. 2, 6 39. 41. No solution 43. 3 45. About44.1 mi 47. About 680 ft 49. About 117 ft 51. About4.3 mi 53. 151.25 ft; 281.25 ft 55. About 25°F57. About 0.81 ft 59. 61. 62. Jeff: Grace: 4 hr 63. 2808 mi 64. 84 hr 65. 0, 66. 0, 67. 68. 69. Left to the student71. 6912 73. 0 75. 77. 2 79. 0, 81. 2 83. 85. 3


1. 6.403 2. 2.449 3. 14.1424. 90.554 ft 5. 7.4 ft


1. J 2. B 3. O 4. M 5. K 6. I 7. G 8. E9. F 10. A


1. 5.831 3. 21.213 5. 5 7. 6.5579. 3.464 11. 13. ft; 10.770 ft15. 7.1 ft 17. ft; 102.767 ft 19.21. cm; 13.454 cm 23. 12 in. 25.27. ft; 26.439 ft 29.31. 33. Flash: Crawler:

34. 35. 36. 3, 8 37. 1 38.39. 13 40. 7 41. 26 packets 43.


1. 2. 3. 4. �6i5. 6. 7. 8.9. 10. 11. 12. 4213. 14. 15.16. 17. 18. 19. 1 20.21. 22. 23. 3 24. 25.26. 27. 28. 29. 53 30. 1031. 32.33.

? 0

TRUE

Yes 0

�1 � 1 i2 � 1

��i�2 � 1

x2 � 1 � 0

1017 �

1117 i2i

�14 i�9 � 5i6 � 3i

�1 � i�7 � 6i8 � i�1i�i5 � 12i11 � 10i

�14 � 8i�35 � 25i�9 � 12i�34�1012 � 7i

3 � 5i�12 � 6i15 � 3i3i6, or 36i�i11, or �11i5ii5, or 5i

75 cm�2, 2�7, 323 3

4 mph

53 23 mph67 2

3 mph;DW420.125 in. 20.497 in.340 � 8

�3, 0�, ��3, 0�181s � s210,561

116n � 112;43;450;34;

8200 ft;200;6;41;

12

1254�6, �3

�3, 72�8, 853

�2.81 1

3 hr;4 49 hrDW

�1

809

164�6

�1925

496

192

A-45

Chapter 10

34.

? 0

TRUE

Yes


1. 2. 3.4. 5. 6. 7.8. 81 9. 10.


1. or 3. 5. or 7. or 9. 11. or 13.15. or 17.19. 21. 23. 25.27. 29. 31. 33.35. 21 37. 39. 41.43. 45. 47. 49.51. 53. 55. 1 57. 59.61. 63. 65. 8 67. 69. 0

71. 0 73. 1 75. 77. 79.

81. 83. 85. 87.89. 91. 93.95.

? 0

TRUE

Yes97.

? 0

FALSE

No99. 101. Rational 102. Difference of squares103. Coordinates 104. Positive 105. Proportion106. Trinomial square 107. Negative 108. Zeroproducts 109. 111.113. 115. 8 117. 119. 1


1. True 2. False 3. False 4. True 5. False6. True


1. 27.893 2. 6.378 3. and do not exist as real numbers; 4. Domain or

5. 6. 7. 8. �x � 3��c � 3�7�z�9�a��163 , ��

� �x � x �163 �,f �41

3 � � 5f �1�f �0�, f ��1�,

35 �

95 i�88i

�3 � 4i�4 � 8i; �2 � 4i; 8 � 6i

DW

�10 4 � 4i � 1 � 8 � 4i � 5

4 � 4i � i2 � 8 � 4i � 5 �2 � i�2 � 4�2 � i� � 5

x2 � 4x � 5 � 0

0 1 � 4i � 4 � 2 � 4i � 5

1 � 4i � 4i2 � 2 � 4i � 5 �1 � 2i�2 � 2�1 � 2i� � 5

x2 � 2x � 5 � 0

35 �

45 i�

12 �

14 i�

43 i

�8

41 �1041 i6

5 �25 i�

37 �

87 i�i

910 �

1310 i2 �

62

i5 � 8i

1 � 23i�125i�1i�1�i�5 � 12i

�24 � 10i5 � 12i2 � 46i38 � 9i18 � 14i1 � 5i�6 � 24i�14�1811 � 6i�1 � i

�4 � 4i7 � 4i9 � 5i12 � 4i�2 � 23 �i4 � 2i154 � 215i,

�7i72i7i2,9i3ii3,�23i�2i3,4i35ii35,

�14

169 �34

169 i117 � 118i�

1625 �

150 i�28.373�20�

151290 �

73290 i

�47 � 161i20 � 17i�2 � 9i

0 1 � 2i � 1 � 2 � 2i � 2

1 � 2i � i2 � 2 � 2i � 2 �1 � i�2 � 2�1 � i� � 2

x2 � 2x � 2 � 0

anspgs041-050 1/19/06 2:53 PM Page A-45

ISB

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-536

-477

42-6


9. 10. 11. 12. 13. 314. 15. 512 16. 17. 18.

19. 20. 21. 22.

23. 24. 25. 26. 27.28. 29. 30. 31.

32. 33. 34. 35.

36. 37. 38. 39. 40.41. 42. 43.44. 45. 46.

47. 48. 49. 13 50. 4 51. 4

52. 9 cm 53. 4.899 ft 54. About 4166 rpm55. 4480 rpm 56. 25 57. 6.78258. 59. 60. 61. 2962. 63. 64. 65. 3 66. No67.

68. The procedure for solving radical equations is toisolate one of the radical terms, use the principle of powers,repeat these steps if necessary until all radicals areeliminated, solve, and then check the possible solutions. A check is necessary since the principle of powers does notalways yield equivalent equations. 69. 70. 3


1. [10.1a] 12.166 2. [10.1a] 2; does not exist as a realnumber 3. [10.1a] Domain or 4. [10.1b] 5. [10.1b] 6. [10.1c] 7. [10.1d] 8. [10.1d] 4 9. [10.2a] 10. [10.2a] 8 11. [10.2a] 12. [10.2a]

13. [10.2b] 14. [10.2b] 15. [10.2c]

16. [10.2c] 17. [10.2d] 18. [10.2d]

19. [10.2d] 20. [10.2d] 21. [10.3a]

22. [10.3a] 23. [10.3a]

24. [10.3b] 25. [10.3b]

26. [10.3a] 27. [10.3a] 28. [10.3b] 29. [10.3b] 30. [10.4a] 31. [10.4b] 32. [10.4b]

33. [10.5b] 34. [10.6a] 35 35. [10.6b] 7

36. [10.6a] 5 37. [10.7a] 7 ft 38. [10.6c] 3600 ft39. [10.7a] 9.899 40. [10.7a] 2 41. [10.8a] 11i98;

13 � 82�41

9 � 6x � x�203822a5 x2y2

xy4 x3 10xy2

5x6y2

2x3 2x2y2

y3

2a3b43 3a2b24 5

237128y7

15a6b5

2xx4 x1

2.931/24

x8/5

y9/58a3/4

b3/2c2/51

10

�5xy2�5/2371/23 a2x

�1

10�x � 5�3�q��, 2�� x � x � 2�,

�1

DW

x

y

f (x) � œx

25 �

35 i9 � 12ii

1 � i�2 � 9i�5 � 22 �i46;

24 ft;

2a � 2ba � b

263

9 � 3 48 � 27�43 � 210152�2x � y2�3 x33

73 x12x55

2xy3 615a5b9

3a3 a2b215xy2x2

3y343 x2

765b23 2a2�33 475

12x7

12x4y33x2x771/6

1x2/5

3at 1/4

5b1/2

a3/4c2/31

4x2/3y2/3

17�a2b3�1/5311/25 a

�x�2; �2; 3�13�10

A-46

Answers

42. [10.8b] 43. [10.8c] 44. [10.8d] 45. [10.8e] 46. [10.8f] No 47. [10.8c, e] 48. [10.6b] 3

CHAPTER 11


1. (a) (b) 2, 4; (c) The solutions of2 and 4, are the first coordinates of the

x-intercepts, and of the graph of2. (a) 4 and or

(b) 3. (a) 0, (b)4. (a) (b) 5. and or

1.732 and or 6. and or

1.633 and or 7. and

or 8. (a) (b)

9. 10. 11.

12. 13. 14.

15. 16. About 33.5 in. 17. 1 sec


Left to the student


The calculator returns an ERROR message because the graphof has no x-intercepts. This indicates that theequation has no real-number solutions.


1. (a) or (b) 3. (a)

or (b) no x-intercepts 5.

7. 9. 8, 0 11. 13.646, 8.354

13. 15. 18, 0 17. 3.371,

19. 5, 21. 9, 5 23. 25.

27. 29. 31.

33. 35. 37.

39. 41. (a) (b)

43. (a) (b) no x-intercepts

45. 47. 49.

51. 53. 55. 0.866 sec2 � 3i�

12

� i

72

23

�7

334

�145

434

�17

4

54

� i39

4;��

72

�57

2, 0�

��

72

�57

2, 0�,�

72

�57

2;5 � 29

�11 � 192, �894

�105

4

�

34

�57

452

�53

2�

12

�5

2

11 � 233�2 � 6�11

�0.37132

�14

2;7 � 2i

11 � 7;5, �9

�1.225�

62

;� 53 i;�

53 i,

53 i,��5, 0�, �5, 0��5;5, �5,

4x2 � 9 � 0y � 4x2 � 9

13

� i

23

13

�22

3�

32

�19

2�3 � 10

10, �2�2, �4�8 � 11

�1 � 5, 0�, �1 � 5, 0�1 � 5;�

22

i

�

22

i,2

2i�1.633�1.633,�

263

;

�

263

,26

3�1.732�1.732,

�3;�3,3�35 , 0�, �1, 0�3

5 , 1;��

72 , 0�, �0, 0��

72 ;��4, 0�, �4, 0�

�4;�4,f �x� � x2 � 6x � 8.�4, 0�,�2, 0�

x2 � 6x � 8 � 0,�2, 0�, �4, 0�;

�174 i�

7750 �

725 i

�i37 � 9i7 � 5i

anspgs041-050 1/19/06 2:53 PM Page A-46

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57. About 8.1 sec 59. About 6.8 sec 61.63. (a) where t is the number ofyears since 1995; (b) 680,000 people; (c) 202064. 65.

66. 67.

68. 69. 70. 4 71. 5 72. 4

73. No solution 75. Left to the student 77.79.


1. (a) (b) 2.

3. or 4.




1. The graph has no x-intercepts. 2. Yes; 2 3. Yes; 24. Yes; 1


1. 2. 3. 3, 8 4. 2, 4


1. 3. 5. 7.

9. 11. 13. (a) 0,

(b) 15. (a) (b)

17. (a) (b) 19. �1, �2�25 , 0�2

5 ;�3 � 22922

, 0��3 � 229

22, 0�,

3 � 22922

;�0, 0�, ��1, 0�

�1;�1 � 2i�3 � 41

2

2 � 3i12

� i

32

�4 � 133

�4 � 14

�1.5, 5�3, 0.8

5 � 736

; 2.257, �0.59112

� i7

21 � i7

2,

�1 � 223

; 1.230, �1.897� 12 , 4�

12 , 4;

0, 72 , �8, � 103

16, �16

105

222

x

y

f (x) � �5 � 2x�x

y

2x � 5y � 10

x

y

f (x) � 5 � 2x

x

y

f (x) � 5 � 2x 2

T �t� � 33.75t � 241.25,

DW

A-47

Chapters 10–11

21. 5, 10 23. 25. 27.

29. 31. 33.

35. 37.

39. 5.236, 0.764 41. 2.766,

43. 1.914, 45. 47. 2 48. 3

49. 10 50. 8 51. No solution 52. No solution53. 54. 55. Left to the student; 0.570

57. 59.

61. 63.


1. 10 ft 2. Distance d is 12 ft; distance to top of ladder is16 ft. 3. Distance d is about 4.782 ft; distance to top ofladder is about 8.782 ft.4.

20 knots, 30 knots

5. 6. 7.

8.


1. B 2. G 3. F 4. L 5. N 6. C 7. J 8. E9. K 10. A


1. Length: 9 ft; width: 2 ft 3. Length: 18 yd; width: 9 yd5. Height: 16 m; base: 7 m 7. Height: ft; base:

ft 9. 3 cm 11. 6 ft, 8 ft 13. 28 and 2915. Length: width:

17.

19.21. First part: 60 mph; second part: 50 mph23. 40 mph 25. Cessna: 150 mph; Beechcraft: 200 mph;or Cessna: 200 mph; Beechcraft: 250 mph 27. ToHillsboro: 10 mph; return trip: 4 mph 29. About 11 mph

31. 33. 35.

37. 39. k �3 � 9 � 8N

2b � c2 � a2

c � Em

r � Gm1m2

Fs � A

6

239 � 7 ft 8.460 ft7 � 239 ft 22.460 ft;

17 � 1092

in. 3.280 in.

14 � 2 ft 1.742 ft2 � 14 ft 5.742 ft;

82 � 44 � 82

b �ta

1 � t 2

t ��g � g 2 � 64s

32r � V

hw2 �

w1

A2

3 � 13�1 � 35

6

�i � i1 � 4i2

1 � 1 � 854

�0.797,173

154

DW�0.3144 � 31

5;

�1.2663 � 65

4;3 � 5;

�3 � 5; �0.764, �5.2361, �12

� i3

2

12 �

32 i3

4 , �22 � 10

23 , 322 � i

17 � 24910

DISTANCE SPEED TIME

FasterShip 3000

SlowerShip 3000 r t

t � 50r � 10

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42-6


A-48

Answers

41. 43.

45. 47. 49.

51. 52. 53.

54. 55. 56. 57.

59. 61. 63.


1. Two real 2. One real 3. Two nonreal4. 5.6. 7. 8.9. 4 10. 11.


1. One real 3. Two nonreal 5. Two real 7. One real9. Two nonreal 11. Two real 13. Two real 15. Onereal 17. 19.21. 23.25.27. 29. 31. 1, 8133. 35. 37. 1 39. 1, 4, 6

41. �1, �3 43. 45.

47. 49. 51.

53. 55. 57.

59. 61. Kenyan:

30 lb; Peruvian: 20 lb 62. Solution A: 4 L; solution B: 8 L63. 64. 65. 66. 467. 68.

69. 70.

x

y

f (x) � �x � 3

x

y

y � 4

x

y

5x � 2y � 8

x

y

f (x) � �Ex � 4

3a4 2a3x24x

DW�3 � 332

, 0��3 � 332

, 0�,

�4, 0�, ��1, 0�,� 425 , 0�9 � 89

2, �1 � 3

�32�

116 , � 1

6�1, 125

�15

3, �

62

�1, 2

�1,�14 , 19�1, 1, 5, 7

�3x2 � 3x � 6 � 012x2 � �4k � 3m�x � km � 0

25x2 � 20x � 12 � 0x2 � 16x � 64 � 0x2 � 9x � 14 � 0x2 � 16 � 0

��3, 0�, ��1, 0�, �2, 0�, �4, 0��13 , 12

�3, �1x2 � 2x � 4 � 0x2 � 25 � 03x2 � 7x � 20 � 0x2 � 5x � 14 � 0

l �w � w5

2A�S � �

S6

�2

4ba�3b2 � 4a�

3x � 33x � 1

2i53x2x

�x�x � 3� �x � 1�

x3 � x2 � 2x � 2�x � 1� �x2 � x � 1�

1x � 2

DWv �cm2 � m2

0

mH � 704.5W

I

g �42L

T 2r ��h � 2h2 � 2A

2

71. Left to the student 73. (a) (b)75. 77.79. 81. 259 83. 1, 3


10099

a � 1, b � 2, c � �3x2 � 3x � 8 � 0�

13�

35 ;

1.

x

y

8

8

f (x) � �ax 2

2.

x

y

8

8

f (x) � 3x 2

3.

x

y

8

8

f (x) � �2x 2

4.

x

y

8

8Vertex:(4, 0)

f (x) � q(x � 4)2

x � 4

5.

x

y

8

8

Vertex:(4, 0)

f (x) � �q(x � 4)2

x � 4

6.

x

y

8

8

Vertex:(�2, �4)Minimum: �4

x � �2

f (x) � q(x � 2)2 � 4

7.


Left to the student


1. For the graph opens up. For the graph ofis narrower than the graph of For

the graph of is wider than the graph of y � x2.y � ax20 � a � 1,y � x2.y � ax2

a � 1,a � 0,

f (x) � �2(x � 5)2 � 3

x

y

8

8

Vertex:(5, 3)

x � 5

Maximum: 3

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2. For the graph opens down. For the graphof is narrower than the graph of For

the graph of is wider than the graph of


Left to the student


If h is positive, the graph of is the graph ofshifted h units to the right. If h is negative, the

graph of is the graph of shifted units to the left.


If k is positive, the graph of is the graphof shifted up k units. If k is negative, thegraph of is the graph of shifted down units.


1.

3.

x

y

f (x) � ax 2

x � 0

Vertex:(0, 0)

x

y

f (x) � 4x 2

x � 0

Vertex:(0, 0)

�k�y � a�x � h�2y � a�x � h�2 � k

y � a�x � h�2y � a�x � h�2 � k

�h�y � ax2y � a�x � h�2y � ax2

y � a�x � h�2

y � x2.y � ax2�1 � a � 0,

y � x2.y � ax2a � �1,a � 0,

A-49

Chapter 11

0 01 42 16

�1 4�2 16

x f �x�

0 01

2

�1

�2 43

13

43

13

x f �x�

5.

x

y

Vertex:(0, 0)

f (x) � �qx 2

x � 0

7.

x

y

Vertex:(0, 0)

f (x) � �4x 2

x � 0

9.

x

y

Vertex:(�3, 0)

f (x) � (x � 3)2

x � �3

11.

x

y

f (x) � 2(x � 4)2

x � 4

Vertex:(4, 0)

13.

x

y

Vertex:(�2, 0)

f (x) � �2(x � 2)2

x � �2

15.

x

y

Vertex:(1, 0)

f (x) � 3(x � 1)2

x � 1

17.

x

y

Vertex:(�2, 0)

f (x) � �w(x � 2)2

x � �2

19.

x

y

Vertex:(3, 1)

Minimum: 1

x � 3

f (x) � (x � 3)2 � 1

21.

x

y

Vertex:(�4, 1)

Maximum: 1

x � �4

f (x) � �3(x � 4)2 � 1

23.

x

y

Minimum: 4

f (x) � q(x � 1)2 � 4

Vertex:(�1, 4)

x � �1

25.

x

y

Maximum: �2

f (x) � �(x � 1)2 � 2

Vertex:(�1, �2)

x � �1

�3 0�2 1�1 4�4 1�5 4

x f �x�

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42-6


27. 29. 9 30. 31. No solution 32. 1533. 34. 35. Left to the student


1.

2.

3. 4. 5.

6. 7. y-intercept: x-intercepts: 8. y-intercept: x-intercept: 9. y-intercept:

; x-intercepts: , , or


1. F 2. H 3. A 4. I 5. C 6. J 7. G 8. B9. E 10. D


�0.268, 0��3.732, 0�,�2 � 3, 0��2 � 3, 0��0, 1�

��4, 0��0, 16�;��3, 0�, �1, 0��0, �3�;�3

2 , 4�

�4,�5��3, �5�

x

y

8

8

f (x) � �4x 2 � 12x � 5

x � w

Vertex:(w, 4)

Maximum: 4

� �4(x � w)2 � 4

x

y

Vertex:(4, �5)

x � 4

Minimum: �5

� 3(x � 4)2 � 5f (x) � 3x 2 � 24x � 43

8

8

x � 2

f (x) � x 2 � 4x � 7

x

y

Vertex:(2, 3)

Minimum: 3

� (x � 2)2 � 3

12a2b25xy24 x

113

DW

A-50

Answers

1.

x

y

(1, �4)

x � 1

f (x) � x 2 � 2x � 3

Minimum: �4

3.

x

y

Maximum: 2 (�2, 2)

f (x) � �x 2 � 4x � 2

x � �2

5.

x

y

(4, 2)

f (x) � 3x 2 � 24x � 50

x � 4

Minimum: 2

7.

x

y

Maximum: r (�q, r)

f (x) � �2x 2 � 2x � 3

x � �q

9.

x

y

(0, 5)

f (x) � 5 � x 2

x � 0

Maximum: 5

11.

x

y

x � �@

f (x) � 2x 2 � 5x � 2

41(�@, �‹) Minimum: �‹41

13. y-intercept: x-intercepts: 15. y-intercept: x-intercepts:17. y-intercept: x-intercept:

19. y-intercept: x-intercepts: none 21.23. (a) (b) 630 mg 24. (a)

(b) 712 calories 25. 250; 26. 250;

27. 28. 29. (a) Minimum:(b) maximum: 7.014�6.954;

2125 ; y � 2

125 x1252 ; y � 125

2 x

y �250

xy �

250x

C � 896 t;D � 15w ;

DW�0, 8�;��

32 , 0��0, 9�;�5, 0�, ��4, 0�

�0, 20�;�3 � 22, 0��3 � 22, 0�,�0, 1�;

31.

x

y

f (x) � �x 2 � 1�

33.

x

y

f (x) � �x 2 � 3x � 4�

35.37. 39.


1. 25 yd by 25 yd 2.3. 4. Polynomial, neither quadratic norlinear 5. Polynomial, neither quadratic nor linear

f �x� � mx � bf �x� � ax2 � bx � c, a � 0

�0, 2�, �5, 7�

x

y

(�1, �q) Minimum: �q

x � �1

f (x) � x 2

�‹ x) � ≈

f �x� � 516 x2 �

158 x �

3516

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6. (a) ; (b) yes;

(c) (d) $912.50


1. Minimum: 1 2. Minimum: 4.875 3. Maximum: 64. Maximum: 0.5625


1. (a)

(b)(c) (d) 15.4 ft; the estimates of 15.4 ft and

15 ft are relatively close.


1. 180 ft by 180 ft 3. 3.5 in. 5. 3.5 hundred, or 3507. 10 ft by 20 ft 9. 11 days after the concert wasannounced; about 62 11.$237,100 at 13. 121; 11 and 11 15. 2 and

17. 36; and 19.21. 23. Polynomial, neitherquadratic nor linear 25.27. 29.31. (a) (b) about 531 per200,000,000 kilometers driven33.35. 37. Radical; radicand 38. Dependent39. Sum 40. At least one 41. Inverse42. Independent 43. Descending 44. x-intercept45. ,where x is the number of years after 1997


1. or 2. or 3. or

4. or 5. or 6.or 7. or 8. or ��, 5� � �10, ��x � x � 5 or x � 10�,

�2, 72��x � 2 � x �

72�,��, �1� � �0, 1�

�x � x � �1 or 0 � x � 1�,��4, 1��x � �4 � x � 1�,��, �4� � �1, ��x � x � �4 or x � 1�,��3, 1�

�x � �3 � x � 1�,��3, 1��x � �3 � x � 1�,��, �3� � �1, ��x � x � �3 or x � 1�,

y � �0.290x4 � 2.699x3 � 8.306x2 � 9.190x � 12.235

DWh�d � � �

1147 d2 �

67 d � �0.0068d2 � 0.8571d

A�s� � 316 s2 �

1354 s � 1750;

f �x� � �14 x2 � 3x � 5f �x� � 2x2 � 3x � 1

a � 0f �x� � ax2 � bx � c,f �x� � ax2 � bx � c, a � 0

f �x� � mx � b�6�6�2�4;x � 490

P�x� � �x2 � 980x � 3000;200 ft2;

y � �0.0082833093x2 � 0.8242996891x � 0.2121786608;

f �x� � �0.02x2 � 9x � 100;

200

400

600

800

1000

x

P

1200

$1400

�200400 500300200100

Days

Pro

fit

A-51

Chapter 11


1. or 2. or 3. or 4. or


1. or 3. or 5. or

7. or 9. All realnumbers, or 11. or 13. or 15. or 17. or 19. or 21. or

23. or 25. or 27.or 29. or 31. or 33.or 35. or 37. or

39. 41. 42.

43. 44. 45. 46.

47. 48. 49. Left to thestudent 51. or

53. All real numbers except 0, or 55. or 57. (a) or

(b) or


1. False 2. True 3. False 4. True 5. False6. True


1. (a) (b) 2. 0,

3. 3, 9 4. 5. 6. 3, 5

7. 8. 4, 9.

10. 11. 12. 0.901 sec

13. Length: 14 cm; width: 9 cm 14. 1 in. 15. First part:50 mph; second part: 40 mph 16. Two real 17. Twononreal 18. 19.

20. 21. 22. 2, �2, 3, �3T � 3B2A

p �9�2

N 2

x2 � 8x � 16 � 025x2 � 10x � 3 � 0

�2 � 31 � 481

15

4 � 42�2�2 � 3; �0.268, �3.732

72

� i3

2�

38

� i7

8

�5

14��14

2, 0�, �14

2, 0��

142

;

�10, ��t � t � 10�,�0, 2�;�t � 0 � t � 2�,��, 1

4� � �52 , ��

�x � x �14 or x �

52�,��, 0� � �0, ��

�1 � 3, 1 � 3 ��x � 1 � 3 � x � 1 � 3 �,

310 � 45�10a � 7�3 2a

17523c7d

3 c24ab2 a

52a

53

DW��, �3� � ��2, 1� � �4, ��

�x � x � �3 or �2 � x � 1 or x � 4�,�0, 1

3��x � 0 � x �13�,��, �4� � �1, 3�

�x � x � �4 or 1 � x � 3�,�1, 2��x � 1 � x � 2�,��3, 0��x � �3 � x � 0�,��, �1� � �2, 5�

�x � x � �1 or 2 � x � 5�,�2, 52��x � 2 � x �

52�,

��23 , 3��x � � 2

3 � x � 3�,��, �1� � �3, ��x � x � �1 or x � 3�,��, 6��x � x � 6�,

��, �3� � ��2, 1��x � x � �3 or �2 � x � 1�,��9, �1� � �4, ��x � �9 � x � �1 or x � 4�,

��, �2� � �0, 2��x � x � �2 or 0 � x � 2�,�2, 4��x � 2 � x � 4�,��, ��

��1, 2��x � �1 � x � 2�,��1, 4��x � �1 � x � 4�,��2, 2��x � �2 � x � 2�,

��, �2� � �6, ��x � x � �2 or x � 6�,

��4, 0� � �4, ��x � �4 � x � 0 or x 4�,��, �2� � �0, 0.5��x � x � �2 or 0 � x � 0.5�,

��2, 3��x � �2 � x � 3�,��, �4� � �1, ��x � x � �4 or x � 1�,

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42-6


23. 3, 24. 25. 81, 16 26. (a)(b) (c) maximum: 3; (d)

27. (a) (b) (c) minimum: (d)

28. (a) (b) (c) maximum: 4; (d)

29. y-intercept: x-intercepts: 30. 11 and 31.32. (a) (b) about105 33. or 34. or

35. The graph of is a parabola with

vertex The line of symmetry is

If then the parabola opens down and is the

maximum function value. If the parabola opens up

and is the minimum function value. The

x-intercepts are and

for When

there is just one x-intercept,

When there are no x-intercepts.36. The x-coordinate of the maximum or minimumDW

b2 � 4ac � 0,

��b

2a, 0�.b2 � 4ac � 0,

b2 � 4ac � 0.��b � b2 � 4ac2a

, 0�,

��b � b2 � 4ac2a

, 0�4ac � b2

4a

a � 0,

4ac � b2

4aa � 0,

x � �b

2a.��

b2a

,4ac � b2

4a �.

f �x� � ax2 � bx � cDW��, �4� � ��2, 1��x � x � �4 or �2 � x � 1�,

��2, 1� � �2, ��x � �2 � x � 1 or x � 2�,N�x� � �0.720x2 � 38.211x � 393.127;

f �x� � �x2 � 6x � 2�11�121;�2, 0�, �7, 0��0, 14�;

x �2

x

y

(�2, 4)

f (x) � �3x 2 � 12x � 8

Maximum: 4

x � �2;��2, 4�;

x

y

x q

(q, _)23

f (x) � x 2 � x � 6

Minimum: _23

234 ;x � 1

2 ;�12 , 23

4 �;

x

y

(1, 3)

x 1

f (x) � �q(x � 1)2 � 3

Maximum: 3

x � 1;�1, 3�;�7, �2�5

A-52

Answers

point lies halfway between the x-coordinates of the x-intercepts. The function must be evaluated for this valueof x in order to determine the maximum or minimum value.37. minimum: 38. 39. 18 and 324


1. [11.1a] (a) (b)

2. [11.2a] 3. [11.4c] 49, 1 4. [11.2a] 9, 2

5. [11.4c]

6. [11.2a] 0.449, 7. [11.2a] 0, 2

8. [11.1b] 9. [11.1c] About 6.7 sec10. [11.3a] About 2.89 mph 11. [11.3a] 7 cm by 7 cm12. [11.1c] About 0.866 sec 13. [11.4a] Two nonreal14. [11.4b]

15. [11.3b] or 16. [11.6a] (a)

(b) (c) maximum: 1; (d)

17. [11.6a] (a) (b) (c) minimum: 5; (d)

18. [11.6b] y-intercept: x-intercepts: 19. [11.7a] 4 and 20. [11.7b] 21. [11.7b] (a) (b) about$2617 billion; about $3085 billion22. [11.8a] or 23. [11.8b] or 24. [11.8b] or 25. [11.6a, b], [11.7b] ; maximum:

26. [11.2a] 27. [11.4c] �2, �2, �2i, �2i12

817f �x� � �

47 x2 �

207 x � 8

��3, 1� � �2, ��x � �3 � x � 1 or x 2�,��3, 5��x � �3 � x � 5�,��1, 7��x � �1 � x � 7�,

C�t� � 2.405t2 � 19.05t � 349;f �x� � 1

5 x2 �35 x�4�16;

�2 � 3, 0�, �2 � 3, 0��0, �1�;

x

y

8

8

(3, 5)Minimum: 5

f (x) � 4x 2 � 24x � 41

x 3

x � 3;�3, 5�;

x

y

8(�1, 1)

Maximum: 1f (x) � �x 2� 2x

x �18

x � �1;

��1, 1�;3V

12T � V

48,

x2 � 43x � 9 � 0

2 � 3

�4.449�2 � 6;

�5 � 52

, �5 � 52

�12

� i3

2

�233

, 0�, ��23

3, 0��

233

;

k � 60h � 60,�

11215f �x� � 7

15 x2 �1415 x � 7;

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CHAPTER 12


1. (a) (b)

2. (a) (b)

3. 4.

5. 6.

7.

x

y

8

8

x � 3y

x

y

f (x) � 2x � 4

x

y

f (x) � 2x �2

x

y

f (x) � (~)x

x

y

f (x) � 4x

x

y

f (x) � (a)x

x

y

f (x) � 3x

A-53

Chapters 11–12

8. $42,000; $44,100 9. (a)(b) $40,000, $48,620.25, $59,098.22, $65,155.79; (c) 10. $9925.67




1. $1125.51 2. $1127.16 3. $30,372.654. (a) $10,540; (b) $10,547.29; (c) $10,551.03; (d) $10,554.80;(e) $10,554.84


1.

3. 5.

7. 9.

x

y

f (x) � 2x � 3

x

y

f (x) � 3x�2

x

y

f (x) � 2x �1

x

y

f (x) � 5x

x

y

f (x) � 2x

t

A

A(t) � $40,000(1.05)t

2 4 6 8 10 12

$100,000

90,000

80,000

70,000

60,000

50,000

40,000

30,000

20,000

10,000

A�t� � $40,000�1.05�t;

0 11 32 93 27

�1

�2

�3 127

19

13

x f x�

0 11

2

3�1 3�2 9�3 27

127

19

13

x f x�

0 11 22 43 8

�1

�2

�3 18

14

12

x f x�

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11.

13.

15. 17.

19. 21.

23. 25.

x

y

8

8

y � 2x

x � 2y

x

y

x � 5y

x

y

x � (q)y

x

y

x � 2y

x

y

f (x) � 22x�1

x

y

f (x) � (Q)x

x

y

f (x) � (q)x

x

y

f (x) � 5x�3

A-54

Answers

27. (a) (b) $50,000; $53,000; $56,180;$63,123.85; $79,692.40; $89,542.38; $160,356.77(c)

29. (a) 17.7 million, 20.1 million, 21.2 million, 30.2 million;(b)

31. (a) $9809, $3527, $1909; (b)

33. (a) 4243; 6000; 8485; 12,000; 24,000;

(b) 35. 37.

38. 39. 1 40. 1 41. 42. 2.7 43.

44. 45. 46. 47. or 625

49. 51.

x

y

y � u(q)x � 1u

x

y

y � 2x � 2�x

54,xx1

x10

1x7

23

1x12

1x2

DW

t

N(t)

1000

2000

3000

4000

5000

6000

2 4 6 8 10 12

N(t) � 3000(2)t/20

Minutes

Nu

mb

er o

f bac

teri

a

t

A(t ) � 9809(0.815)t

642

100020003000400050006000700080009000

$10,000

8 10 12

A(t )

Pri

ce

Years since 2000

t

N(t ) � 17.7(1.018)t

10 20 30

50

40

30

20

10

N(t )

Nu

mb

er o

f cas

eso

f dia

bet

es(i

n m

illi

on

s)

Years since 2000

t

A(t ) � $50,000(1.06)t

5 10 15 20 25

$250,000

200,000

150,000

100,000

50,000

A(t )

A�t� � $50,000�1.06�t;

0 11

2

3�1 2�2 4�3 8

18

14

12

x f x�

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53. 55. Left to the student


1. Inverse of

2. Inverse:

3. Inverse:

4. Yes 5. No 6. Yes 7. No 8. (a) Yes; (b) (c)

f �1(x) � 3 � x

x

yf �1�x� � 3 � x;

x

y

y x

y � x 2 � 4x � 7

x � y 2 � 4y � 7

x � y2 � 4y � 7;

x

y

y xx � 6 � 2y

y � 6 � 2x

x � 6 � 2y;

x

y

(2, 5)(�1, 4)

(�2, 1) (5, 2)

(4, �1)(1, �2)

y x

g � ��5, 2�, �4, �1�, �1, �2��

x

y

y x

y � 3�(x�1)

x � 3�(y�1)

A-55

Chapter 12

9. (a) Yes; (b)

(c)

10.

11. (a) Yes; (b)(c) 12.

13. 14. (a)

(b)

15.


1. 2.

�9

�6

9

6

Inv of y

y �x23

�9

�6

9

6

y x � 5

Inv of y

� x � 4 � 4 � x

�23 �3x � 12

2 � � 4 �6x � 24

6� 4

f � f �1�x� � f � f �1�x�� f�3x � 122 �

�2x2

� x;

�3�2

3 x � 4� � 12

2�

2x � 12 � 122

f �1 � f �x� � f �1� f �x�� f �1�23 x � 4�

f �x� �1

x4 , g�x� � x � 5

f �x� � 3 x, g�x� � x2 � 1;43 x � 5; 3 4x � 5

x2 � 4; x2 � 10x � 24

x

y

8

8

f (x) � x 3 � 1

f �1(x) � œx � 1ßßÍ3

y x

f �1�x� � 3 x � 1;

x

y

g(x) � 3x � 2

y x

x � 23g�1(x) �

x

y

g �1(x ) � �� x � 2

3

g�1�x� �x � 2

3;

6 02 20 3

�4 5

x y

7 04 13 24 37 4

x y

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42-6


3. 4.


1. Left to the student 2. 4; 7 3. 9; 9 4. 1; 25


1. Inverse:

3. Inverse:

5. Yes 7. No 9. No 11. Yes 13.

15. 17.

19. 21. Not one-to-one

23. 25.

27.29.

x

y

f �1(x) � 2x � 6

f (x) � qx � 3

f �1�x� � 2x � 6f �1�x� � x3

f �1�x� � 3 x � 1f �1�x� �1 � 3x5x � 2

f �1�x� �2x

� 5

f �1�x� � 34 �x � 7�f �1�x� �

�2x

f �1�x� �x � 2

5

x

y

y xy � 2x � 6

x � 2y � 6

8

8

x � 2y � 6

x

y

(1, 2)

(2, 1)

(6, �3)

y x

(�5, �3)

(�3, 6)

(�3, �5)

��2, 1�, ��3, 6�, ��5, �3��

�9

�6

9

6

Inv of y y x 3 � 3Inv of y

y x 2 � 2�9

�6

9

6

A-56

Answers

31.

33. 35.

37. 39.

41.

43. 45.

47.

49.51.

53.

55. 57. 59.61. (a) 40, 42, 46, 50; (b) (c) 8, 10, 14, 1863. 65. 66. 67. 68.69. 70. 71. 72. 73.74. 75. 76. 77. No 79. Yes81. (1) C; (2) A; (3) B; (4) D 83.

85. yesg�x� � 2x � 6;f �x� � 12 x � 3;

x

y

3pq33a2b210x3y62a3b84 p2t2xy24 23

2t 2a2b33 x23 aDWf �1�x� � x � 32;

f �1�x� � x3 � 5f �1�x� � �xf �1�x� � 13 x

�1 �

1x � 11

x � 1

�

xx � 1

1x � 1

� x

f � f �1�x� � f � f �1�x�� f� 1x � 1�

�1

1 � xx

� 1�

11x

� x;

f �1 � f �x� � f �1� f �x�� f �1�1 � xx �

f � f �1�x� � f � f �1�x�� f �54 x� � 4

5 �54 x� � x

f �1 � f �x� � f �1� f �x�� f �1�45 x� � 5

4 �45 x� � x ;

f �x� � x4, g�x� � x � 5

f �x� �1

x, g�x� � 7x � 2

f �x� �1x

, g�x� � x � 1f �x� � x, g�x� � 5x � 2

f �x� � x2, g�x� � 5 � 3x

x4 � 10x2 � 30; x4 � 10x2 � 2016x2 � 1;

24x2 � 1

12x2 � 12x � 5; 6x2 � 3�8x � 9; �8x � 18

x

y

f (x) � x 3

f �1(x) � œxß3

f �1�x� � 3 x

4 �16 08 1

10 212 3

x y

0 01 12 83 27

�1 �1�2 �8�3 �27

x f x�

0 01 18 2

27 3�1 �1�8 �2

�27 �3

x f �1 x�

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1. is the power to which we raise 2 to get 64; 62. 3.

4.5.6.

7. 8. 9. 10.11. 10,000 12. 3 13. 14. 4 15. 16. 017. 0 18. 1 19. 1 20. 0 21. 4.893422. 23. Does not exist as a real number24. between 3 and 4; 3.994525.

26. 78,234.8042


1.

3. 5.

7. 9. 11.

13. 15. 17.19. 21. 23.25. 27. 29.31. 33. 35. 9 37. 4 39. 441. 3 43. 25 45. 1 47. 49. 2 51. 253. 55. 0 57. 4 59. 2 61. 3 63.65. 0 67. 1 69. 71. 4.8970 73.75. Does not exist as a real number 77. 0.946479.

81.83. Conjugate 84. Direct 85. Coefficient; exponent

DW7000 � 103.845170,000,000 � 107.8451;98,760.6 � 104.9946;0.00987606 � 10�2.0054;

987,606 � 105.9946;84 � 101.9243;6 � 100.7782;

�0.173923

�2�1

12

t k � Qe4.6052 � 100100.9031 � 810�2 � 0.0162 � 36

4w � 10�2 � loge 0.13532 � loge 7.3891t � logQ x2 � loge t0.3010 � log10 2

13 � log8 2�3 � log5

1125

3 � log10 1000

x

y

8

8

f (x) � 3x

f �1(x) � log 3 x

x

y

f (x) � log 1/3 x

x

y

f (x) � log 2 x

x � 2y

1860 � 103.269518,600,000 � 107.2695;0.000778899 � 10�3.1085;0.00708 � 10�2.1500;

634,567 � 105.8025;8 � 100.9031; 947 � 102.9763;log 1000 � 3, log 10,000 � 4;�4.5100

�414

t x � Ma7 � Q103 � 100025 � 32

T � logm P0.25 � log16 2�3 � log10 0.0010 � log6 1

x

y

8

8

y � f (x) � log 3 x

log2 64

A-57

Chapter 12

86. Quadratic; discriminant 87. Inconsistent88. Parabolas 89. Line of symmetry 90.91. 93. 25 95. 32

97. 99. 3 101. 0 103.


1. 2. 3.4. 5. 6.7. 8.9.10.11.

12. 13. or 14. 0.602

15. 1 16. 17. 0.398 18. 19.20. 1.699 21. 1.204 22. 6 23. 3.2 24. 12


1. Not correct 2. Correct 3. Not correct 4. Correct5. Not correct 6. Correct 7. Not correct 8. Notcorrect


1. 3.5. 7. 9.11. 13. 15.17. 19. 21.23.25.27.

29.

31. or 33. 35.

37. 2.708 39. 0.51 41. 43. 45. 2.609

47. 49. 5 51. 7 53. 55. 57.58. 59. 5 60. 61. 62.63. 64. 65. Left to the student67.

69. 71. False 73. True75. False


1. 11.2675 2. 3. Does not exist as a realnumber 4. Does not exist 5. 2.3026 6. 78,237.1596

�10.3848

12 loga �1 � s� �

12 loga �1 � s�

loga �x6 � x4y2 � x2y4 � y6�3 � 4i�34 � 31i

10i23 � 18i35 �

45 i�1

iDW�7t

32�1.609

loga a

xloga

2x4

y3loga 3 x2

yloga

x2/3

y1/2 ,

2 loga m � 3 loga n �34 �

54 loga b

43 logc x � logc y �

23 logc z

logb x � 2 logb y � 3 logb z2 loga x � 3 loga y � loga z

logc 223logb 2 � logb 5loga 67 � loga 5

�3 logb C6 logb t4 logc ylogc Kylogb 252loga Q � loga x

log4 64 � log4 16log2 32 � log2 8

32�0.699�0.398

loga b�3/2loga 1

bb,loga

x5z1/4

y

3 loga x � 4 loga y � 5 loga z � 9 loga w2 loga x � 3 loga y � loga z

32 loga z �

12 loga x �

12 loga y

log2 5logb P � logb Q

12 loga 55 log7 4loga �JAM �

log3 35logb P � logb Qlog5 25 � log5 5

�2�7

16

x

y

f (x) � log 3 ux � 1u

a � bi

1 02 14 28 3

�1

�2

�318

14

12

x, or 2

y y

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A-58

Answers

7. 0.1353 8. (a) 1.0860; (b) 1.0860 9. 5.523610. 11.

12. 13.


1. 2.

3. 4.


1. J 2. B 3. O 4. G 5. N 6. F 7. A 8. H9. I 10. K


1. 0.6931 3. 4.1271 5. 8.3814 7.9. 11. Does not exist 13. 15. 117. 15.0293 19. 0.0305 21. 109.9472 23. 525. 2.5702 27. 6.6439 29. 2.1452 31.33. 35. 4.6284�2.3219

�2.3219

�1.7455�1.6094�5.0832

�1 10

y log2/3 x

�10

10

�1

�6

10

6y log1/2 x

�1

�3

10

4y log3 x

�1

�3

10

6y log2 x

x

y

g(x) � ln (x � 2)

x

y

f (x) � 2 ln x

x

y

g(x) � qe�x

x

y

f (x) � e 2x

37.

39. 41.

43. 45.

47.

49. 51.

x

y

f (x) � 2 ln x

x

y

f (x) � ln (x � 3)

x

y

f (x) � ln (x � 2)

x

y

f (x) � e x � 1

x

y

f (x) � e x�2

x

y

f (x) � e x�1

x

y

f (x) � e�0.5x

x

y

f (x) � e x

0 11 2.72 7.43 20.1

�1 0.4�2 0.1�3 0.05

x f x�

0 0.71 1.12 1.43 1.6

�0.5 0.4�1 0�1.5 �0.7

x f x�

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53. 55.

57. 59. 16, 256 60. 61. 49, 12162. 63. Domain: range: 65. Domain: range: 67.


1. 1 2. 3 3. 1.5395 4. 14.6068 5. 25 6.7. 2 8. 5


Left to the student


1. 3 3. 4 5. 7. 9. 3.4594 11. 5.426313. 15. 17. 19. 4.6052 21. 2.302623. 140.6705 25. 2.7095 27. 3.2220 29. 25631. 33. 10 35. 37.

39. 41. 121 43. 10 45. 47. 3

49. 51. 5 53. No solution 55. 57.

58. 59. 60.

61. 62. 63. 1 65. (a) 0.3770;

(b) (c) 0.9036; (d) 67. 3, 4 69.71. 2 73. 75. 77. 1, 10079. 3, 81. 1, 1.465


1. About 65 decibels 2. 3. About 4.94. moles per liter 5. (a) 43.5 yr; (b) 16.5 yr6. About 300 million; about 344 million 7. (a)

(b) $5309.18; $5637.48; $9110.59; (c) about11.6 yr 8. (a) where is inbillions of dollars and is the number of years after 2000; (b) about $34,531 billion, or $34.531 trillion; (c) 20089. 2972 yr


1.2. 3. 3,531,046 businesses

y � 309,870.3567 � 1.090787419x

tP�t�P�t� � 2.8e0.471t,k � 0.471,

P�t� � 5000e0.06t;k � 6%,

10�710�9.2 W�m2

�710100,000�34

�4�1.5318�1.9617;

�iy4/3

25x2z4

�1

10 , 1�2, �3, �5 � 41

2�64, 8

�10, �2DW25

13

1e

� 0.3679

e2 � 7.38911100

132

32�3, �15

2

35

52

25

�52 , ��, 100��, ��;

�0, ��, ��;�3, �4

14 , 9DW

x

y

f (x) � u ln x u

x

y

f (x) � q ln x � 1

A-59

Chapter 12


1. D 2. M 3. I 4. A 5. E 6. H 7. C 8. G9. N 10. B


1. About 95 dB 3. or about5. About 6.8 7. moles

per liter 9. 11.13. (a) 243 people; (b) about 20.6 months; (c) about 0.6 month 15. (a) $19,796; (b) 2011; (c) 11.9 yr17. (a) (b) 6.9 billion; (c) 2043; (d) 60.8 yr19. (a) (b) $5309.18; $5637.48; $9110.59; (c) in 11.6 yr 21. (a)(b) 2,888,380; (c) 2027 23. About 2103 yr 25. About7.2 days 27. 69.3% per year 29. (a)

(b) 4.7 million tons; (c) 217331. (a) (b) $2,419,866; (c) 9.9 yr; (d) 2002; (e) The function predicts that the card’s value willbe about $908,202 in 2001. According to this, the card wouldnot be a good buy at $1.1 million. 33. (a)

(b) about 2,081,949; (c) 211535. 37. 38. 1 39. i 40. i 41.42. 43. 44. 45. 4146. 47. 1.078, 58.77049. 2, 4 51. $13.4 million


1. False 2. True 3. True 4. False 5. True6. True 7. True 8. True


1. 2. Not one-to-one

3. 4.

5. 6.

7.

x

y

f (x) � 3 x�1

x

y

y xf (x) � x 3 � 1

3f �1(x) � œx � 1

f �1�x� �3x � 4

2x

f �1�x� � 12 3 xg�1�x� �

7x � 32

��2, �4�, ��7, 5�, ��2, �1�, �11, 10��

�0.767,�0.937,91 � 60i

�2

41 �2341 i63

65 �1665 i�2

�1 � i�1DWP�t� � 2,394,811e�0.007t;

k � 0.007,

V�t� � 451,000e0.07t;W�t� � 17.5e�0.094t;

k � 0.094,

P�t� � 852,737e0.061t;k � 0.061,P�t� � P0e0.06t;P�t� � 6.4e0.0114t;

2.17 ft�sec3.04 ft�sec1.58 � 10�83.2 � 10�2 W�m2

10�1.5 W�m2,

01 12 33 9

�1

�2

�3 181

127

19

13

x f x�

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42-6


8.

9.

10.

11.12. answers may vary13. 14. 15.16. 17. 2 18. 19. 1 20. 021. 22. Does not exist as a real number23.

24. 25. 26.

27. 17 28. 29. 8.7601 30. 3.2698 31. 2.5499532. 33. 34. 0.3753 35. 18.356836. 0 37. Does not exist 38. 1 39. 0.4307

40. 1.7097 41. 42. 2 43.

44. 45. 46. 1, 47. 1.5266

48. 35.0656 49. 2 50. 8 51. 52.53. 90 dB 54. (a) 0.807 million; 1.581 million; 45.737 million; (b) in 14.5 yr; (c) 2 yr;

43175

�572e�2 � 0.1353

110,000

19

�2.6921�3.6602�7

log a1/2

bc2loga 12012 log z �

34 log x �

14 log y

4 loga x � 2 loga y � 3 loga z�2.7425

�1�12��3

� 8

4x � 1612 � log25 54 � log 10,000

g�x� � 4 � 7x;f �x� � x,g � f �x� � 3x2 � 5f � g�x� � 9x2 � 30x � 25;

x

y

f (x) � ln (x � 1)

x

y

f (x) � e x�1

x

y

y � log 3 x

x � 3 y

A-60

Answers

(d)

55. (a) (b) $102,399; (c) after 8 yr, or in 2013 56.57. About 8.25 yr 58. About 3463 yr 59. Youcannot take the logarithm of a negative number becauselogarithm bases are positive and there is no real-numberpower to which a positive number can be raised to yield anegative number. 60. because

61. 62.


1. [12.1a] 2. [12.3a]

3. [12.5c] 4. [12.5c]

5. [12.2a]

6. [12.2b, c]

7. [12.2b, c] 8. [12.2b] Not one-to-one9. [12.2d] 10. [12.3b] 11. [12.3b] 12. [12.3c] 3 13. [12.3c] 23 14. [12.3c] 015. [12.3d] 16. [12.3d] Does not exist as a realnumber 17. [12.4d]

18. [12.4d] 19. [12.4d]

20. [12.4d] 1.079 21. [12.5a] 6.693822. [12.5a] 107.7701 23. [12.5a] 0 24. [12.5b] 1.188125. [12.6b] 5 26. [12.6b] 2 27. [12.6b] 10,00028. [12.6b] 29. [12.6a] 0.0937e1/4 � 1.2840

�0.544loga x1/3z2

y3

3 log a �12 log b � 2 log c

�1.9101

7m � 49log256 16 � 12

g � f �x� � 5x2 � 5x � 2f � g�x� � 25x2 � 15x � 2,f �1�x� � 3 x � 1

f �1�x� �x � 3

4

��3, �4�, ��8, 5�, ��3, �1�, �12, 10��

x

y

f (x) � ln (x � 4)

x

y

f (x) � e x�2

x

y

y � log 2 x

x

y

f (x) � 2x�1

�83 , �

23�ee 3

a0 � 1.loga 1 � 0DW

DWk � 0.231

V�t� � 40,000e0.094t;k � 0.094,

t

S(t ) � 0.15(1.4)t

50

60

70

80

40

30

20

10

S(t )

Nu

mb

er o

f su

bsc

rip

tio

ns

(in

mil

lio

ns)

5 2010 15

Years since 2003

1 03 19 2

27 3�1

�2

�3127

19

13

x, or 3

y y

0 2.71 7.42 20.13 54.6

�1 1�2 0.4�3 0.1

x f x�

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30. [12.6b] 9 31. [12.6b] 1 32. [12.7a] 4.233. [12.7b] (a) $31,081; (b) 2010; (c) 15.7 yr34. [12.7b] (a) where t is thenumber of years since 2000 and N is in millions; (b) 33.672 million, 35.862 million; (c) 2052; (d) 77.0 yr35. [12.7b] About 4.6% 36. [12.7b] About 4684 yr37. [12.6b] 44, 38. [12.4d] 2

Cumulative Review/Final Examination, p. 933

1. [1.2e] 2. [4.1e, f ] 3. [1.8d] 62.54. [1.8c] 5. [9.1c] or 6. [9.3e] or 7. [9.2a] or 8. [8.2a], [8.3a] 9. [8.5a]

10. [5.8b] 11. [6.7a] 12. [10.6a] 4

13. [5.8b] 14. [11.2a]

15. [11.4c] 16. [6.7a] 17. [12.6b] 1 18. [12.6a] 1.748 19. [12.6b] 920. [11.8a] or 21. [11.8b] or

22. [2.4b]

23. [12.1c], [12.7b] (a) About 74.97 billion about84.94 billion (b) about 39 yr; (c)

24. [12.7b] (a) (b) $50,000;$58,492.93; $68,428.45; $74,012.21; (c)

25. [4.5d]

26. [4.4b, c] 27. [6.1d]

28. [6.5b] 29. [6.6a]

30. [4.8b]

31. [10.3b] 32. [10.4a] 33. [10.2d] 834. [10.8c] 35. [10.8e] 36. [7.4a], [7.5e] (a) $434.7 billion, $488.7 billion,$524.7 billion;

310 �

1110 i16 � i�2

11�25x2�y

x3 � 2x2 � 2x � 1 �3

x � 1

1x � 1

x � 2x � 1

2m � 1m � 1

�x3 � 3x2 � x � 6

2x3 � x2 � 8x � 3

t

A(t)

A(t) � $50,000(1.04)t

2 4 6 8 10 12

$80,000

70,000

60,000

50,000

40,000

30,000

20,000

10,000

A�t� � $50,000�1.04�t;

t

N(t)

N(t) � 65(1.018)t

2 4 6 8 10 12 14 16

100

90

80

70

60

50

Years since 2000

Am

ou

nt

of l

um

ber

(in

bil

lio

ns

of c

ub

ic fe

et)

ft3;ft3,

N �4P � 3M

6

��, �1� � �2, ��x � x � �1 or x � 2�,��, �1� � �1, ��x � x � �1 or x � 1�,

�16�3, �2, 2, 3

�14

� i�7

4�7, �3

�23 , 3�1, 32

��1, 2, 3��13 , 5�

��1, 6��x � �1 � x � 6�,��, �6.4� � �6.4, ��x � x � �6.4 or x � 6.4�,

��1, ��x � x � �1�,�x � 30�9x6y22

15

�37

N�t� � 31.902e0.009t,k 0.009,

A-61

Chapter 12

(b)

(c) (d) 18; (e) an increase of $18 billion per year37. [7.4a] 38. [9.4b]

39. [9.4c] 40. [11.6a]

41. [12.5c] 42. [12.3a]

43. [5.1c] 44. [5.2a] 45. [5.2a] 46. [5.5d] 47. [5.5b] 48. [5.6a] 49. [5.3a], [5.4a] 50. [5.3a], [5.4a] 51. [7.5c] 52. [7.5d] 53. [12.2c]

54. [6.9e] 55. [7.1b] 56. [10.5b]

57. [7.2a] 58. [7.2a] All real numbers 59. [9.1d] More than 460. [6.8a] 612 mi 61. [2.6a] 62. [8.4a] 24 L of A; 56 L of B 63. [6.8a] 350 mph64. [6.8a] 65. [6.9f] 20 66. [11.7a] 67. [12.7b] 2397 yr 68. [6.9d] 3360 kg69. [7.5c] f �x� � �

13 x �

73

1250 ft28 25 min

11 37

��, �13� � ��

13 , 0� � �0, ��

15 � 8�a � a9 � a

�1045

f �1�x� � 12 �x � 3�y � �

12 x �

52

y � 2x � 23x�2x � 1� �x � 5�2�5x � 2� �x � 7�

3�3a � 2� �9a2 � 6a � 4�4�2x � 1�2�9m2 � n2� �3m � n� �3m � n��x � 8� �x � 9�

3�a � 9b� �a � 5b��2x3 � 1� �x � 6�

x

y

f (x) � log 2 x

x

y

f (x) � e�x

f (x) � 2x 2 � 8x � 9x

y

x

y

y , �2

x

y

4y � 3x � 12

x

y

�0, 344.7�;

t

S

S(t) � 18t � 344.7

0 10 15 205 25

$800

600

400

200

Years since 2000

To

tal s

ales

(in

bil

lio

ns)

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70. [11.7b] 71. [12.3b]

72. [12.3b] 73. [12.4d]

74. [12.4d] 75. [11.7a] 169 76. [12.2b] No77. [7.2a] (a) (b) (c) (d)78. [12.7b] (a) (b) 530,878; (c) about2042

79. [5.8b] 0, 80. [1.8d], [4.1f]

81. [10.6a], [11.4c] or

82. [5.6a]

APPENDIXES

Margin Exercises, Appendix A, pp. 940–943

1. 1, 3, 9 2. 1, 2, 4, 8, 16 3. 1, 2, 3, 4, 6, 8, 12, 244. 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 1805. 13 6. 7. 8.9. 15, 30, 45, 10. 45, 90, 135, 11. 4012. 54 13. 360 14. 18 15. 24 16. 36 17. 210

Exercise Set A, p. 944

1. 1, 2, 4, 5, 10, 20 3. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 725. 7. 9. 11. 13.15. 17. 19.21. 23. 25. 5; 2027. 72 29. 3; 1531. 120 33. 13; 23; 29935. 90 37. 18039. 120 41. 17; 29; 49343. 84 45. 2; 3; 5; 3047. 7249. 5; 60 51. 36

Margin Exercises, Appendix B, pp. 946–952

1. 2. 3. answers may vary 4.

5. 6. 7. 8. 4 9. 10. 11.

12. 2 13. 14. 15. 16. 17.

18. 19. 20. 3 21. 22. 23.

24. 25. 26. 81

Exercise Set B, p. 953

1. 3. 5. 7. 9. 4 11. 13. 8

15. 17. 5 19. 21. 23. 25. 27. 4516

112

43

137

1721

14

17

23

2124

60100

912

164

815

56

821

2120

15

715

114

736

1940

4360

2315

75

3516

52

12

83

199

25

2832

;2124

,1416

,2128

812

2 3 3;2 2 3;2 3;3 5;2 2 3;2 2 3;2 2 3 3;2 2 2 3;

2 2 7;2 2 3;2 3 5;2 2 2 3;

2 2 3 3;2 3 5;2 3 5;2 3 3;2 2 2 5;2 3 5;

3 5;2 2 3 3;2 2 2 3;2 2;7 177 13

2 3 5 72 3 3 52 2 2 52 3 37 73 32 113 5

180, . . .60, . . .2 5 7 112 5 52 2 2 2 3

a2

�2b9 � a2

4�

ab9

�4b2

81��51 � 7�61

1942

51 � 7�61 ,

3283

ba

P�t� � 152,099e0.05t;��7, ��2, �1, 1, 2;��, ��;�5;

�6 logb x � 30 logb y � 6 logb z

logb x7

yz8�1/5

3x � Q

log r � 6f �x� � �1718 x2 �

5918 x �

119

A-62

Answers

29. 31. 33. 35. 37. 39. 41.43. 500

Margin Exercises, Appendix C, pp. 954–956

1. 2. 3. 4. 10,000 5. 512 6. 1.3317. 5 8. 14 9. 13 10. 1000 11. 250 12. 178

13. 2 14. 125 15. 48 16.

Exercise Set C, p. 957

1. 3. 5. 7. 49 9. 59,049 11. 100

13. 1 15. 5.29 17. 0.008 19. 416.16 21.

23. 125 25. 1061.208 27. 25 29. 114 31. 3333. 5 35. 12 37. 324 39. 100 41. 1000 43. 2245. 1 47. 4 49. 102 51. 96 53. 24 55. 90

57. 8 59. 1 61. 50,000 63. 5 65. 27 67.

69. 71. 9

Margin Exercises, Appendix D, pp. 959–965

1. 2.3. 4. , or 5. (a) ; (b) The product of each pair ispositive. 6. 7.8. 9.10. 11.12. 13.14. 15. 16.17. 18.19. 20.21. 22.

Exercise Set D, p. 966

1. 3.5. 7.9. 11.13. 15.17. 19.21. 23.25. 27.29. 31. 33.35. 37. 39.41. 43. 45.47. 49.51. 53.55. 57.59. 61.63. 3�2a � 1� �4a2 � 2a � 1�

2� y � 4� � y2 � 4y � 16��2 � 3b� �4 � 6b � 9b2��2a � 1� �4a2 � 2a � 1��x � 1� �x2 � x � 1�

�z � 3� �z2 � 3z � 9��16 � z� �1

6 � z�a�2a � 7� �2a � 7�� p � 7� � p � 7�

�x � 4� �x � 4�3�2a � 3�2y� y � 9�2�3y � 2�2�x � 1�2� y � 9�2

�x � 2�2�3x � 1� �3x � 4��3a � 4� �a � 1�x2�7x � 1� �2x � 3�x�3x � 4� �4x � 5�

4�2x � 1� �x � 4�2�5t � 3� �t � 1��5y � 2� �7y � 4��3c � 8� �c � 4�y�5y � 7� �2y � 3��3x � 1� �x � 5�

�p �15� �p �

15�� y � 6� � y � 4�

�m � 9� �m � 8�2a�a � 5� �a � 5��t � 5� �t � 3�� y � 7� � y � 3�

�2 � 4y� �4 � 8y � 16y2��x � 10� �x2 � 10x � 100��1 � x� �1 � x � x2��w � 3� �w2 � 3w � 9�

�m �13�

�m �

13�� y � 2� � y � 2�

�3y � 5�2�x � 7�22x3�2x � 3� �5x � 4�2�4y � 1� �3y � 5��3x � 2� �x � 1�

�x � 7� �3x � 8��x � 10� �x � 11�� y � 6� � y � 2�x�x � 6� �x � 2�

2x�x � 7� �x � 6�x�x � 9� �x � 6��x � 5� �x � 4�

�3 � t� �8 � t��t � 3� �t � 8��m � 2� �m � 6�� y � 2� � y � 5��x � 2� �x � 3�

1966

2245

964

10610354

112

�1.08�26543

103

3518

3160

1324

12

56

76

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Margin Exercises, Appendix F, pp. 970–971

1. 2. 3. 16 yd 4.

5. 66 man-hours 6. 7. 84 in. 8.

Exercise Set F, p. 972

1. 68 ft 3. 45 g 5. 7. 9.

11. 12 yd 13. 15. 17. 51.2 oz

19. 21. 96 in. 23. Approximately 0.03 yr

25. Approximately 31,710 yr 27. 29. 172,800 sec

31. 33.

Margin Exercises, Appendix G, pp. 973–974

1. 56.7 2. 64.7 3. 87.8 4. 17 5. 16.5 6. 917. 55 8. 54, 87 9. No mode exists. 10. (a)(b) (c) No mode exists.

Exercise Set G, p. 975

1. Mean: 21; median: 18.5; mode: 29 3. Mean: 21;median: 20; modes: 5, 20 5. Mean: 5.2; median: 5.7;mode: 7.4 7. Mean: 239.5; median: 234; mode: 2349. Mean: median: 15; mode: 1 11. Mean: 897.2;median: 798; no mode exists

Margin Exercises, Appendix H, p. 977

1. R 34; or

2. R or

3.

Exercise Set H, p. 978

1. R or

3. R or

5. R or

7. R or

9. R 12; or

11. 13.15. y3 � 2y2 � 4y � 8

x2 � 2x � 43x3 � 9x2 � 2x � 6

y2 � 2y � 1 �12

y � 2y2 � 2y � 1,

3x2 � 2x � 2 ��3

x � 3�3;3x2 � 2x � 2,

x2 � 5x � 23 ��43

x � 2�43;x2 � 5x � 23,

a � 7 ��47

a � 4�47;a � 7,

x2 � x � 1 ��4

x � 1�4;x2 � x � 1,

y2 � y � 1

x2 � 4x � 13 ��30

x � 2�30;x2 � 4x � 13,

2x2 � 2x � 14 �34

x � 32x2 � 2x � 14,

23.8;

25 mm2;25 mm2;

1.08 tonyd3

32 ft2

80 ozin.

3080 ft�min

$970day

16 ft3

4 in.-lb

sec3.3

msec

15 mihr

132 ft

sec800

kW-hrda

72 in225 m

sec70

mihr

A-63

Chapters 12–Appendixes

Margin Exercises, Appendix I, pp. 979–982

1. 2. 3. 4. 93 5.6.

Exercise Set I, p. 983

1. 10 3. 0 5. 7. 0 9. 11.13. 5 15. 0 17. 19.21. 23. 25. 27.

29. 31.

Margin Exercises, Appendix J, pp. 985–986

1. 2.

Exercise Set J, p. 988

1. 3. 5. 7.

9. 11. 13.

15. 17.

Margin Exercises, Appendix K, p. 990

1. (a) (b) 6; (c) (d) (e)

2. (a) (b) 15; (c) (d) 6

Exercise Set K, p. 991

1. 1 3. 5. 12 7. 9. 5 11. 213. 15. 21 17. 519. 21. 42 23. 25.

27.

29.

31.

Margin Exercises, Appendix L, pp. 993–995

1. 10 2. 3. 4.5.

6. 7. �x � 3�2 � � y � 1�2 � 36�0, 0�; r � 8

x

y

Center: (5, �q)Radius: 3

8

10

�9, �5��32 , �3��37 6.083

3�4 � x�5�x � 2�

15�x � 2� �4 � x�

;3

x � 2�

54 � x

;3

x � 2�

54 � x

;

14x3�x � 2�

4x3

x � 2;

1x � 2

� 4x3;1

x � 2� 4x3;

x2

3x � 43x3 � 4x2;x2 � 3x � 4;x2 � 3x � 4;

16�

34�x3 � 4x2 � 3x � 12

x2 � x � 1

1318�41

�54;�5;

x4 � 6x2 � 9x2 � 3x2 � 3

;x4 � 9;2x2;

�w, x, y, z� � �1, �3, �2, �1��4, 12 , �

12�

�0, 2, 1��2, �2, 1��32 , �4, 3�

�10, �10��12 , 3

2��4, 3��32 , 5

2�

��1, 2, 3��8, 2�

�2, �2, 1��32 , �4, 3�

�1, 2, 3��2, �1, 4��4, 3��32 , 5

2��25

2 , �112 ��2, 0�

�3�10�48

�1, 3, �2��3, �1��6�7�5

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8. Center: radius:

Exercise Set L, p. 996

1. 5 3. 5. 7. 7.1

9. 11. 13.

15.

17. 19. 21.

23. 25. 27.

29. 31.

x

y

Center: (�1, �3)Radius: 2

(x � 1)2 � (y � 3)2 � 4

�2 � �32

,32�

�� 112 , 1

24��0.25, �0.3��1, �172 �

�0, 112 ��3

2 , 72��9,672,400 3110.048

�17 � 2�14 � 2�15 5.677

�a2 � b2�6970 83.487�41

7 0.915

�648 25.456�29 5.385

x

y

x 2 � 2x � y 2 � 4y � 2 � 0

�3;��1, 2�;

A-64

Answers

33. 35.

37. 39.41. 43.45. �2, 0�, r � 2

�4, �1�, r � 2��4, 3�, r � 2�10�x � 5�2 � � y � 3�2 � 7x2 � y2 � 49

x

y

Center: (0, 0)Radius: 5

x 2 � y 2 � 25

x

y

(x � 3)2 � y 2 � 2

Center: (3, 0)Radius: �2

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G-1

GLOSSARY

Glossary

AAbscissa The first coordinate in an ordered pair of

numbersAbsolute value The distance that a number is from 0 on

the number lineac-method A method for factoring trinomials of the type

involving the product, ac, of theleading coefficient a and the last term c

Additive identity The number 0Additive inverse A number’s opposite; two numbers are

additive inverses of each other if their sum is 0Algebraic expression An expression consisting of

variables, constants, numerals, and operation signsArea The number of square units that fill a plane regionArithmetic numbers The whole numbers and the positive

fractionsAscending order When a polynomial is written with the

terms arranged according to degree from least to greatest, it is said to be in ascending order.

Associative law of addition The statement that whenthree numbers are added, regrouping the addends givesthe same sum

Associative law of multiplication The statement thatwhen three numbers are multiplied, regrouping the factors gives the same product

Asymptote A line that a graph approaches more and moreclosely as x increases or as x decreases

Average A center point of a set of numbers found byadding the numbers and dividing by the number ofitems of data; also called the arithmetic mean or mean

Axes Two perpendicular number lines used to identifypoints in a plane

Axis of symmetry A line that can be drawn through agraph such that the part of the graph on one side of the line is an exact reflection of the part on the opposite side

BBar graph A graphic display of data using bars

proportional in length to the numbers representedBase In exponential notation, the number being raised to

a power

Binomial A polynomial composed of two termsBreak-even point In business, the point of intersection of

the revenue function and the cost function

CCircle A set of points in a plane that are a fixed distance r,

called the radius, from a fixed point called thecenter

Circumference The distance around a circleCoefficient The numerical multiplier of a variableCommon logarithm A logarithm with base 10Commutative law of addition The statement that when

two numbers are added, changing the order in whichthe numbers are added does not affect the sum

Commutative law of multiplication The statement thatwhen two numbers are multiplied, changing the orderin which the numbers are multiplied does not affect theproduct

Completing the square Adding a particular constant to an expression so that the resulting sum is a perfectsquare

Complex fraction expression A rational expression thathas one or more rational expressions within its numerator and/or denominator

Complex number Any number that can be written aswhere a and b are real numbers

Complex rational expression A rational expression thathas one or more rational expressions within its numerator and/or denominator

Complex-number system A number system that containsthe real-number system and is designed so that negative numbers have square roots

Composite function A function in which a quantity depends on a variable that, in turn, depends on another variable

Composite number A natural number, other than 1, thatis not prime

Compound inequality A statement in which two or moreinequalities are combined using the word and or theword or

Compound interest Interest computed on the sum of anoriginal principal and the interest previously accruedby that principal

a � bi,

�h, k�,a � 1,ax2 � bx � c,

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Conic section A curve formed by the intersection of aplane and a cone

Conjugates Pairs of radical terms, like and or and for which the

product does not have a radical termConjunction A sentence in which two statements are

joined by the word andConsecutive even integers Even integers that are two

units apartConsecutive integers Integers that are one unit apartConsecutive odd integers Odd integers that are two units

apartConsistent system of equations A system of equations

that has at least one solutionConstant A known numberConstant function A function given by an equation of the

form where b is a real numberConstant of proportionality The constant in an equation

of direct or inverse variationCoordinates The numbers in an ordered pairCube root The number c is called a cube root of a if

DDegree of a polynomial The degree of the term of highest

degree in a polynomialDegree of a term The sum of the exponents of the

variablesDemand function A function modeling the relationship

between the price of a good and the quantity of thatgood demanded

Denominator The number below the fraction bar in afraction

Dependent equations The equations in a system are dependent if one equation can be removed withoutchanging the solution set.

Descending order When a polynomial is written with theterms arranged according to degree from greatest toleast, it is said to be in descending order.

Determinant The determinant of a two-by-two matrix

is denoted and represents

Diameter A segment that passes through the center of acircle and has its endpoints on the circle

Difference of cubes Any expression that can be written inthe form

Difference of squares Any expression that can be writtenin the form

Direct variation A situation that translates to an equationof the form with k a positive constant

Discriminant The radicand, from the quadraticformula

Disjoint sets Two sets with an empty intersectionDisjunction A sentence in which two statements are

joined by the word orDistributive law of multiplication over addition The

statement that multiplying a factor by the sum of twonumbers gives the same result as multiplying the factorby each of the two numbers and then adding

Distributive law of multiplication over subtraction Thestatement that multiplying a factor by the difference oftwo numbers gives the same result as multiplying thefactor by each of the two numbers and then subtracting

Domain The set of all first coordinates of the ordered pairsin a function

Doubling time The time necessary for a population todouble in size

EElimination method An algebraic method that uses the

addition principle to solve a system of equationsEllipse The set of all points in a plane for which the sum

of the distances from two fixed points and is constant

Empty set The set without membersEquation A number sentence that says that the

expressions on either side of the equals sign, �, represent the same number

Equation of direct variation An equation, described bywith k a positive constant, used to represent

direct variationEquation of inverse variation An equation, described by

with k a positive constant, used to represent inverse variation

Equilibrium point The point of intersection between thedemand function and the supply function

Equivalent equations Equations with the same solutionset

Equivalent expressions Expressions that have the samevalue for all allowable replacements

Equivalent inequalities Inequalities that have the samesolution set

Evaluate To substitute a value for each occurrence of avariable in an expression

Exponent In expressions of the form the number n isan exponent. For n a natural number, represents nfactors of a.

Exponential decay A decrease in quantity over time thatcan be modeled by an exponential equation of the form

Exponential equation An equation in which a variable appears as an exponent

Exponential function A function that can be described byan exponential equation

Exponential growth An increase in quantity over time thatcan be modeled by an exponential function of the form

Exponential notation A representation of a number usinga base raised to a power

FFactor Verb: To write an equivalent expression that is a

product. Noun: A multiplierFactorization of a polynomial An expression that names

the polynomial as a productFixed costs In business, costs that are incurred whether or

not a product is produced

k � 0P�t� � P0ekt,

k � 0P �t� � P0e�kt,

anan,

y � k�x

y � kx

F2F1

b2 � 4ac,y � kx,

a2 � b2

a3 � b3

ad � bc.�ab cd

��ab

cd�

c3 � a.

f �x� � b,

c � �d,c � �d�a � �b�a � �b

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GLOSSARY

Focus One of two fixed points that determine the points ofan ellipse

FOIL To multiply two binomials by multiplying the Firstterms, the Outside terms, the Inside terms, and thenthe Last terms

Formula An equation that uses numbers or letters to rep-resent a relationship between two or more quantities

Fraction equation An equation containing one or morerational expressions; also called a rational equation

Fraction expression A quotient, or ratio, of polynomials;also called a rational expression

Fraction notation A number written using a numeratorand a denominator

Function A correspondence that assigns to each memberof a set called the domain exactly one member of a setcalled the range

GGrade The measure of a road’s steepnessGraph A picture or diagram of the data in a table; a line,

curve, or collection of points that represents all the solutions of an equation

Greatest common factor (GCF) The common factor of apolynomial with the largest possible coefficient and thelargest possible exponent(s)

HHalf-life The amount of time necessary for half of a

quantity to decayHypotenuse In a right triangle, the side opposite the right

angle

IIdentity Property of 1 The statement that the product of a

number and 1 is always the original numberIdentity Property of 0 The statement that the sum of a

number and 0 is always the original numberImaginary number A number that can be named bi,

where b is some real number and Imaginary number i The square root of �1; that is,

and Inconsistent system of equations A system of equations

for which there is no solutionIndependent equations Equations that are not dependentIndex In the radical the number n is called the index.Inequality A mathematical sentence using �, �, �, �, or Input A member of the domain of a functionIntegers The whole numbers and their oppositesIntercept The point at which a graph intersects the x- or

y-axisIntersection of two sets The set of all elements that are

common to both setsInterval notation The use of a pair of numbers inside

parentheses and brackets to represent the set of allnumbers between those two numbers

Inverse relation The relation formed by interchanging themembers of the domain and the range of a relation

Inverse variation A situation that translates to an equa-tion of the form with k a positive constant

Irrational number A real number that cannot be namedas a ratio of two integers

JJoint variation A situation that translates to an equation

of the form with k a constant

LLeading coefficient The coefficient of the term of highest

degree in a polynomialLeading term The term of highest degree in a polynomialLeast common denominator (LCD) The least common

multiple of the denominatorsLegs In a right triangle, the two sides that form the right

angleLike terms Terms that have exactly the same variable

factorsLine of symmetry A line that can be drawn through a

graph such that the part of the graph on one side of theline is an exact reflection of the part on the oppositeside

Linear equation Any equation that can be written in theform or where x and y are variables

Linear function A function that can be described by anequation of the form where x and y are variables

Linear inequality An inequality whose related equation isa linear equation

Linear programming A branch of mathematics involvinggraphs of inequalities and their constraints

Logarithmic equation An equation containing a logarith-mic expression

Logarithmic function, base a The inverse of an exponen-tial function with base a

MMaximum value The largest function value (output)

achieved by a functionMean A center point of a set of numbers found by adding

the numbers and dividing by the number of items ofdata; also called the arithmetic mean or average

Median In a set of data listed in order from smallest tolargest, the middle number if there is an odd number ofdata items, or the average of the two middle numbers ifthere is an even number of data items

Minimum value The smallest function value (output)achieved by a function

Mode The number or numbers that occur most often in aset of data

Monomial A constant, a variable, or a product of a constant and one or more variables

Motion problem A problem that deals with distance,speed, and time

y � mx � b,

Ax � Bx � C,y � mx � b

y � kxz,

y � k�x,

��

na,

i2 � �1i � ��1

b � 0


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Multiple of a number A product of the number and somenatural number

Multiplication property of 0 The statement that the product of 0 and any real number is 0

Multiplicative identity The number 1Multiplicative inverses Reciprocals; two numbers whose

product is 1

NNatural logarithm A logarithm with base eNatural numbers The counting numbers: 1, 2, 3, 4, 5, …Nonlinear function A function whose graph is not a

straight lineNumerator The number above the fraction bar in a

fraction

OOne-to-one function A function for which different inputs

have different outputsOpposite The opposite, or additive inverse, of a number a

is written �a. Opposites are the same distance from 0on the number line but on different sides of 0.

Opposite of a polynomial To find the opposite of a polynomial, replace each term with its opposite—that is, change the sign of every term.

Ordered pair A pair of numbers of the form forwhich the order in which the numbers are listed is important

Ordinate The second coordinate in an ordered pair ofnumbers

Origin The point on a graph where the two axes intersectOutput A member of the range of a function

PParabola A graph of a quadratic equationParallel lines Lines in the same plane that never intersect;

two lines are parallel if they have the same slope.Parallelogram A four-sided polygon with two pairs of

parallel sidesPercent notation A representation of a number as parts

per 100Perfect square A rational number p for which there exists

a number a for which Perfect-square trinomial A trinomial that is the square of

a binomialPerimeter The sum of the lengths of the sides of a polygonPerpendicular lines Lines that form a right anglePi (�) The number that results when the circumference of

a circle is divided by its diameter; or 22�7Point–slope equation An equation of the type

where x and y are variablesPolygon A closed geometric figure with three or more

sidesPolynomial A monomial or sum of monomialsPolynomial equation An equation in which two

polynomials are set equal to each other

Polynomial inequality An inequality that is equivalent toan inequality with a polynomial as one side and 0 asthe other

Prime factorization A factorization of a composite num-ber as a product of prime numbers

Prime number A natural number that has exactly two different factors: itself and 1

Prime polynomial A polynomial that cannot be factoredusing only integer coefficients

Principal square root The nonnegative square root of anumber

Principle of zero products The statement that an equation is true if and only if is true or

is true, or both are trueProportion An equation stating that two ratios are equalProportional numbers Two pairs of numbers having the

same ratioPythagorean theorem In any right triangle, if a and b

are the lengths of the legs and c is the length of the hypotenuse, then

QQuadrants The four regions into which the axes divide a

planeQuadratic equation An equation of the form

where Quadratic formula The solutions of

are given by the equation

Quadratic function A second-degree polynomial functionin one variable

Quadratic inequality A second-degree polynomial inequality in one variable

RRadical equation An equation in which a variable appears

in a radicandRadical expression An algebraic expression in which a

radical symbol appearsRadical symbol The symbol Radicand The expression under the radical symbolRadius A segment with one endpoint on the center of a

circle and the other endpoint on the circleRange The set of all second coordinates of the ordered

pairs in a functionRatio The quotient of two quantitiesRational equation An equation containing one or more

rational expressionsRational expression A quotient of two polynomialsRational inequality An inequality containing a rational

expressionRational number A number that can be written in the

form a�b, where a and b are integers and Rationalizing the denominator A procedure for finding

an equivalent expression without a radical in the denominator

Real numbers All rational and irrational numbers

b � 0

�

x ��b �b2 � 4ac

2a .a � 0,

ax2 � bx � c � 0,a � 0ax2 � bx � c � 0,

a2 � b2 � c2.

b � 0a � 0ab � 0

y � y1 � m�x � x1�,

� 3.14,

a2 � p

�h, k�

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Reciprocal A multiplicative inverse; two numbers are reciprocals if their product is 1.

Rectangle A four-sided polygon with four right anglesReflection The mirror image of a graphRelation A correspondence between a first set, the

domain, and a second set, the range, such that eachmember of the domain corresponds to at least onemember of the range

Repeating decimal A decimal in which a number patternrepeats indefinitely

Right triangle A triangle that includes a right angleRise The change in the second coordinate between two

points on a lineRoster notation A way of naming sets by listing all the

elements in the setRounding Approximating the value of a number; used

when estimatingRun The change in the first coordinate between two

points on a line

SScientific notation A representation of a number of the

form where n is an integer, andM is expressed in decimal notation

Set A collection of objectsSet-builder notation The naming of a set by describing

basic characteristics of the elements in the setSimilar triangles Triangles in which corresponding sides

are proportionalSimplify To rewrite an expression in an equivalent,

abbreviated, formSlope The ratio of the rise to the run for any two points on

a lineSlope–intercept equation An equation of the form

where x and y are variablesSolution A replacement or substitution that makes an

equation or inequality trueSolution set The set of all solutions of an equation, an

inequality, or a system of equations or inequalitiesSolve To find all solutions of an equation, an inequality,

or a system of equations or inequalities; to find the solution(s) of a problem

Speed The ratio of distance traveled to the time requiredto travel that distance

Square A four-sided polygon with four right angles and allsides of equal length

Square of a number A number multiplied by itselfSquare root The number c is a square root of a if Substitute To replace a variable with a numberSubstitution method An algebraic method for solving

systems of equationsSum of cubes An expression that can be written in the

form

Sum of squares An expression that can be written in theform

Supply function A function modeling the relationship be-tween the price of a good and the quantity of that goodsupplied

System of equations A set of two or more equations thatare to be solved simultaneously

TTerm A number, a variable, or a product or a quotient of

numbers and/or variablesTerminating decimal A decimal that can be written using

a finite number of decimal placesTotal cost The amount spent to produce a productTotal profit The amount taken in less the amount spent,

or total revenue minus total costTotal revenue The amount taken in from the sale of a

productTrinomial A polynomial that is composed of three termsTrinomial square The square of a binomial expressed as

three terms

UUnion of sets A and B The set of all elements belonging to

either A or B

VValue The numerical result after a number has been

substituted into an expressionVariable A letter that represents an unknown numberVariable expression An expression containing a variableVariation constant The constant in an equation of direct

or inverse variationVertex The point at which the graph of a quadratic

equation crosses its axis of symmetryVertical-line test The statement that a graph represents a

function if it is impossible to draw a vertical line thatintersects the graph more than once

WWhole numbers The natural numbers and 0: 0, 1, 2, 3, …

Xx-intercept The point at which a graph crosses the x-axis

Yy-intercept The point at which a graph crosses the y-axis

a2 � b2

a3 � b3

c2 � a.

y � mx � b,

1 � M � 10,M � 10n,

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Index

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INDEX

AAbscissa, 166Absolute value, 17, 18, 642

and distance, 643equations with, 643, 645inequalities with, 646properties of, 642, 669and radical expressions, 681, 682,

683, 684Absolute-value function, 821Absolute-value principle, 644, 669ac-method, 336Acidosis, 913Addition

associative law, 56, 76commutative law, 55, 76of complex numbers, 737of exponents, 224, 227, 239using fraction notation, 949of functions, 989of logarithms, 889, 926on number line, 22of polynomials, 259, 288of radical expressions, 706of rational expressions, 414–417of real numbers, 22, 23, 49

Addition principlefor equations, 83, 160for inequalities, 141, 160, 617, 669

Additive identity, 53, 945Additive inverse, 24, 25, 260. See also

Opposites.Advance of a pipe, 734Algebra of functions, 989Algebraic equation, 2, 3Algebraic expression, 2

evaluating, 3, 4, 223least common multiple, 411translating to, 5

Algebraic–graphical connection, 178,246, 247, 370, 438, 553, 719,754, 769, 905

Alkalosis, 921And, 631, 632

Anglescomplementary, 562supplementary, 562

Annually compounded interest, 852,926

Approximating roots, 14, 685Approximating solutions, quadratic

equations, 770Area

rectangle, 3right circular cylinder, 291

Arithmetic numbers, 945Ascending order, 251Ask mode, 101Associative laws, 56, 76Asymptote of exponential function,

848At least, 151, 620At most, 151, 620Auto mode, 175Average, 973Axes, 166Axis of symmetry, 797. See also Line of

symmetry.

BBack-substitution, 985Bar graph, 166Base

of an exponential function, 847in exponential notation, 222, 954of a logarithmic function, 879

changing, 896Base-10 logarithms, 883Binomials, 252

difference of squares, 344as a divisor, 296product of, 268

FOIL method, 274, 302squares of, 276, 277, 302sum and difference of terms, 275,

276, 302Body mass index, 627Boyle’s law, 471

Braces, 68Brackets, 68Break-even analysis, 602Break-even point, 603

CCalculator. See also Graphing

calculator.approximating square roots on, 14cube roots on, 685logarithms on, 883square roots on, 685

Calculator Corner, 4, 13, 14, 17, 49, 70,101, 175, 180, 188, 203, 204, 237,248, 262, 270, 332, 371, 400, 440,485, 488, 502, 506, 514, 520, 534,550, 558, 567, 662, 685, 692, 719,741, 758, 770, 772, 773, 797, 798,799, 801, 820, 825, 833, 850, 854,868, 872, 890, 897, 899, 906, 918,996. See also Graphingcalculator.

Canceling, 398Carbon dating, 917Center of a circle, 994Center point of data, 973Central tendency, measure of, 973Change, rate of, 201, 509. See also

Slope.Change-of-base formula, 896Changing the sign, 66, 67Changing units, 971Check the solution. See also Checking.

in equation solving, 84, 101, 367,439, 718

of an inequality, 141, 617in problem solving, 122

Checking. See also Check the solution.addition, 262division, 298factorizations, 317, 332multiplication, 270, 400simplifications, 400subtraction, 262

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INDEX

Circleequation of, 994graphing, 994

Clearing decimals, 96, 98Clearing fractions, 96, 97, 436, 438Coefficients, 249

leading, 317Collecting like terms, 61, 250, 287

in equation solving, 95Columns of a matrix, 984Combining like terms, see Collecting

like termsCommon denominators, 949. See also

Least common denominator.Common factor, 311Common logarithms, 883Common multiples, 942. See also Least

common multiple.Commutative laws, 55, 76Complementary angles, 562Completing the square, 759

in the equation of a circle, 995in graphing, 808

Complex conjugates, 739, 748Complex fraction expression, 430Complex numbers, 736, 737

addition, 737conjugate, 739, 748division, 740on a graphing calculator, 741multiplication, 737as solutions of equations, 742subtraction, 737

Complex rational expression, 430Composite function, 869, 870, 926

and inverses, 871Composite number, 941Compound inequalities, 630Compound interest, 852–854, 926Conjugate

of a complex number, 739, 748of a radical expression, 707

Conjunction, 631Consecutive integers, 125Consistent system of equations, 551,

552, 566Constant, 2

of proportionality, 462, 464variation, 462, 464

Constant function, 485Continuously compounded interest,

914, 926Converting

decimal notation to scientificnotation, 235

fraction notation to decimalnotation, 13

scientific notation to decimalnotation, 236

Coordinates, 166finding, 168

Correspondence, 482Cost, 602Cramer’s rule, 979, 980, 982Cross products, 450Cube, surface area, 110Cube roots, 682Cube-root function, 683Cubes, factoring sum and difference

of, 352, 354, 964, 965Cubic function, 821Cylinder, right circular, surface area,

291

DDecay model, exponential, 917, 926Decay rate, 917Decimal notation. See also Decimals.

converting to/from scientificnotation, 235, 236

for irrational numbers, 15for rational numbers, 13, 14, 15repeating, 14terminating, 14

Decimals, clearing, 96, 98 Degrees of polynomials and terms,

251, 287Demand, 604, 629Denominator, 945

least common, 949rationalizing, 712, 714

Dependent equations, 552, 566, 982Descending order, 250Determinants, 979

and solving systems of equations,980

Dewpoint spread, 629Difference, 30. See also Subtraction.

of cubes, factoring, 352, 354, 964,965

of functions, 989of logarithms, 896, 926of squares, 344

factoring, 345, 354, 388, 964, 965Dimension symbols

calculating with, 970unit changes, 971

Direct variation, 462Directly proportional, 462Discriminant, 788, 840Disjoint sets, 633Disjunction, 634Distance

on the number line, 643in the plane, 992

Distance (traveled) formula, 107, 579.See also Motion formula;Motion problems.

Distributive laws, 57, 58, 76Dividend, 297

of zero, 45

Divisionchecking, 298of complex numbers, 940and dividend of zero, 45using exponents, 225, 227, 239using fraction notation, 951, 952of functions, 989of integers, 44of polynomials, 295–298with radical expressions, 700of rational expressions, 405of real numbers, 44–49and reciprocals, 47, 405, 945using scientific notation, 237synthetic, 976by zero, 44, 45

Divisor, 297Domain

of a function, 482, 483, 497, 680of a relation, 483

Doubling time, 913Draw feature on a graphing calculator,

868

Ee, 895, 926Earned run average, 471Elements of a matrix, 984Elimination method, 563, 567Empty set, 633Endpoints of an interval, 615Entries of a matrix, 984Equation, 82. See also Formulas.

with absolute value, 643, 645addition principle, 83, 160algebraic, 2of a circle, 994of direct variation, 462, 463equivalent, 83exponential, 880, 904false, 82fraction, 436 entering on a graphing calculator, 101graphs, 169. See also Graphs.with infinitely many solutions, 99of inverse variation, 464linear, 173, 587logarithmic, 880, 906multiplication principle, 88, 160with no solution, 100containing parentheses, 98point–slope, 526, 527, 540polynomial, 246quadratic, 366quadratic in form, 790radical, 717rational, 436reducible to quadratic, 790related, 655reversing, 90


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INDEX

slope–intercept, 508, 540solutions of, 82, 168solving, see Solving equationssystems of, 548. See also Systems of

equations.translating to, 122true, 82

Equilibrium point, 605Equivalent equations, 83Equivalent expressions, 53, 395, 945

for fractions, 53for one, 946

Equivalent inequalities, 140, 616Evaluating determinants, 979Evaluating expressions

algebraic, 3, 4, 223exponential, 954polynomials, 245, 248, 286

Evaluating formulas, 106Evaluating functions, 484, 485Even integers, consecutive, 125Even root, 684Exponent, 954. See also Exponential

notation; Exponents.Exponential decay model, 917, 926Exponential equality, principle of, 904Exponential equations, 880, 904. See

also Exponential functions.converting to logarithmic equations,

880solving, 904

Exponential expressions, 954Exponential functions, 847, 926. See

also Exponential equations.graphs of, 846, 898

asymptotes, 848y-intercept, 849

inverse of, 879Exponential growth model, 914, 926Exponential notation, 222, 954. See

also Exponents.Exponential regression, 918Exponents, 222, 954

dividing using, 225, 227, 239evaluating expressions with, 223on a graphing calculator, 692irrational, 846laws of, 692and logarithms, 879multiplying, 232, 239multiplying using, 224, 227, 239negative, 226, 227, 239of one, 223, 227, 239 raising a power to a power, 232, 239raising a product to a power, 233,

239raising a quotient to a power, 234,

239rational, 690, 691rules for, 239of zero, 223, 227, 239

Expressionsalgebraic, 2, 3equivalent, 53, 395, 945

for one, 946evaluating, 3, 4, 954exponential, 954factoring, 59. See also Factoring.fraction, see Rational expressions radical, 679rational, see Rational expressionssimplifying, see Simplifyingterms, 58value of, 3

Extraneous solution, 718

FFactor, 309, 940. See also Factoring;

Factorization; Factors.greatest common, 308–310

Factor tree, 941Factoring, 59, 940

checking, 317, 332common factors, 311completely, 347difference of cubes, 352, 354, 964,

965difference of squares, 345, 354, 388,

964, 965finding LCM by, 942by FOIL method, 961general strategy, 357by grouping, 313 numbers, 940polynomials, 309, 313

with a common factor, 311differences of cubes, 352, 354, 964,

965differences of squares, 345, 354,

388, 964, 965general strategy, 357by grouping, 313sums of cubes, 352, 354, 964, 965tips for, 347trinomial squares, 343, 388, 963,

965trinomials, 317–322, 327–331, 336,

343, 959–963, 965radical expressions, 698solving equations by, 368strategy, 357sum of cubes, 352, 354, 964, 965sum of squares, 347, 354, 965by trial and error, 317trinomial squares, 343, 388, 963, 965trinomials, 317–322, 327–331, 336,

343, 959–963, 965Factorization, 308, 309, 940. See also

Factoring.prime, 941

Factors, 308, 940. See also Factor;Factoring.

and sums, 285Falling object, distance traveled, 253False equation, 82False inequality, 139, 641Familiarization in problem solving,

122First coordinate, 166Fitting a function to data, 534, 821,

825, 918Five-step process for problem solving,

122Fixed cost, 602FOIL method

of factoring, 327, 961of multiplication, 274, 302

Formula, 106body mass index, 627change-of-base, 896compound interest, 854, 926distance between points, 992distance traveled, 107, 579earned run average, 471evaluating, 106 for inverses of functions, 864midpoint, 993motion, 448, 579quadratic, 768, 840simple interest, 131solving, 107–109, 781

Fraction baras a division symbol, 3as a grouping symbol, 956

Fraction equations, 436Fraction expressions, 394. See also

Fraction notation; Rationalexpressions.

addition of, 949complex, 430division of, 951, 952equivalent, 946, 947multiplication of, 948multiplying by one, 395, 949simplifying, 946subtraction of, 950

Fraction notation, 945addition using, 949converting to decimal notation, 13division using, 951, 952equivalent, 53multiplication using, 948sign changes in, 48simplest, 946simplifying, 946subtraction using, 950

Fractions, clearing, 96, 97, 436, 438Functions, 482, 483, 497

absolute-value, 821addition of, 989algebra of, 989


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Functions (continued)composite, 869, 870, 926constant, 485cube-root, 683cubic, 821difference of, 989division of, 989domain, 482, 483, 497, 680evaluating, 484, 485exponential, 847, 926fitting to data, 534, 821, 825, 918graphs of, 486, 488horizontal-line test, 863input, 484inverse of, 864library of, 821linear, 502logarithmic, 879, 895multiplication of, 989notation, 484one-to-one, 862, 863output, 484product of, 989quadratic, 797, 808, 821quartic, 821quotient of, 989range, 482, 483, 497square-root, 680subtraction of, 989sum of, 989value of, 484, 485vertical-line test, 487

GGames in a league, 383Geometry and polynomials, 261Golden rectangle, 787Grade, 200, 508Graph

of absolute-value functions, 821bar, 166circle, 994of cube-root function, 683of cubic functions, 821of direct-variation equations, 463of equations, 169. See also

Graphing.systems of, 549–553

of exponential functions, 846, 898of functions, 486, 488

and their inverses, 867, 868vertical-line test, 487

on a graphing calculator, 180of horizontal lines, 516, 517, 540of inequalities, 139, 653, 655, 662

systems of, 657, 662using intercepts, 513of inverse-variation equations, 465of inverses of functions, 867, 868

of linear equations, 173, 190, 518 on a graphing calculator, 180horizontal line, 188, 189using intercepts, 186, 513slope, 199using slope and y-intercept, 515vertical line, 189x-intercept, 186y-intercept, 175, 186

of linear functions, 821of logarithmic functions, 878, 898,

899nonlinear, 179of numbers, 13of parabolas, 797–803, 808–811, 821of points in a plane, 166of quadratic functions, 797–803,

808–811of quartic functions, 821of radical functions, 680slope, 505, 508, 540

graphing using, 515and solving equations, 553

systems of, 549, 552of square-root functions, 680translations, 503, 800, 849, 850vertical-line test, 487of vertical lines, 517, 540x-intercept, 513y-intercept, 504, 513

Graphing, see GraphGraphing calculator

and absolute value, 17approximating solutions of

quadratic equations, 770and change-of-base formula, 897checking addition and subtraction,

262 checking factorizations, 332checking multiplication, 270, 400checking simplification of rational

expressions, 400checking solutions of equations,

101, 440and complex numbers, 741and composition of functions, 872and compound-interest formula,

854converting fraction notation to

decimal notation, 13and cube roots, 685draw feature, 868entering equations, 101evaluating algebraic expressions, 4evaluating polynomials, 248and exponential functions, 850and exponential notation, 692exponential regression, 918and function values, 485graphing equations, 180, 188

graphing functions, 488graphing inequalities, 662 GraphStyle, 662and grouping symbols, 70and inequalities, 662Inequalz feature, 662and intercepts, 188, 514intersect feature, 550and inverses of functions, 868linear regression, 534and logarithmic functions, 899and logarithms, 883and maximum and minimum

values, 820and negative numbers, 13and operations with real numbers,

49and order of operations, 70and perpendicular lines, 520pi key, 14and polynomial inequalities, 833and powers, 692quadratic regression, 825and rational exponents, 692regression, 534, 825, 918roots, approximating, 683and scientific notation, 237shading, 662slope, visualizing, 203, 204and solutions of equations, 175and solving equations, 719, 773

quadratic, 371systems of, 550

and solving inequalities, 833and square roots, 14, 683squaring the window, 520stat feature, 534tables

ask mode, 101, 485auto mode, 175

trace, 485value feature, 485vars, 535viewing window, 180, 188zero feature, 371

Graphs, see GraphGraphStyle, 662Greater than (�), 15Greater than or equal to (�), 17Greatest common factor (GCF),

308–310Grouping

in addition, 56, 76factoring by, 313in multiplication, 56, 76symbols, 68, 956

Grouping (ac-) method for factoring,336

Growth model, exponential, 914, 926Growth rate, exponential, 914

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HHalf-life, 917Half-plane, 653Handshakes, number possible, 383Hang time, 762Height of a projectile, 839Home-run differential, 36Hooke’s law, 469Horizon, sighting to, 726Horizontal line, 188, 189, 210, 516,

517, 540slope, 204, 210

Horizontal-line test, 863Hypotenuse, 378

Ii, 736, 748

powers of, 739Identity

additive, 53, 945multiplicative, 53, 945

Identity property of one, 53, 76, 945Identity property of zero, 53, 76, 945Imaginary numbers, 736, 748Improper symbol for fractions, 949Inconsistent system of equations, 551,

552, 566, 982Independent equations, 552, 566Index of a radical expression, 683Inequalities, 15, 139, 614

with absolute value, 646addition principle for, 141, 160, 617,

669with “and,” 631, 632checking solutions, 617compound, 630conjunction, 631disjunction, 634equivalent, 140, 616false, 139, 614graphs of, 141, 614, 653, 655, 662

systems of, 657, 662linear, 653multiplication principle for, 143,

160, 618, 669with “or,” 634, 635polynomial, 832–836quadratic, 832rational, 836solution set, 139, 614solutions of, 139, 614, 653solving, 614. See also Solving

inequalities.systems of, 657translating to, 151true, 139, 614

Inequalz feature on a graphingcalculator, 662

Infinitely many solutions, 99Infinity (�), 615

Input, 484Integers, 10

consecutive, 125division of, 44negative, 11positive, 11

Intercepts of exponential functions, 849and graphing calculators, 188, 514graphing using, 513of parabolas, 754, 812x-, 186, 513y-, 175, 186, 210, 504, 513, 812, 849

Interestcompound, 852–854, 914, 916simple, 131

Intersect feature on a graphingcalculator, 550

Intersection of sets, 630, 669Interval notation, 615, 616Introductory algebra review, 968, 969Inverse properties, 76Inverse relation, 860, 861. See also

Inverses of functions.Inverse variation, 464Inversely proportional, 464Inverses

additive, 24, 25, 260. See alsoOpposites.

of functions, 864and composition, 871exponential, 879logarithmic, 884, 896

multiplicative, 45, 951. See alsoReciprocals.

of relations, 860, 861Irrational numbers, 14

as exponents, 846

JJoint variation, 467

Kkth roots

of quotients, 701, 748simplifying, 698

LLaw(s)

associative, 56, 76commutative, 55, 76distributive, 57, 58, 76of exponents, 692

LCD, see Least common denominatorLCM, see Least common multipleLeading coefficient, 317Least common denominator, (LCD),

370, 414, 949

Least common multiple, (LCM), 410,942, 943

of an algebraic expression, 411and clearing fractions, 436

Legs of a right triangle, 378Less than (�), 15Less than or equal to (�), 17Library of functions, 821Like radicals, 706Like terms, 61, 249, 287Line. See also Lines.

horizontal, 188, 189, 210slope, 198vertical, 189, 210

Line of symmetry, 797, 811Linear equations in three variables,

587Linear equations in two variables,

173. See also Linear functions.applications, 178graphs of, 173, 186, 190, 518point–slope, 526, 527, 540slope–intercept, 508, 540systems of, 548

Linear functions, 502, 821. See alsoLinear equations.

Linear inequality, 653. See alsoInequalities.

Linear programming, 660Linear regression, 535Lines

equations of, see Linear equationshorizontal, 516, 517, 540parallel, 518, 540perpendicular, 519, 540slope of, 505, 508, 540vertical, 517, 540

ln, see Natural logarithmslog, see Common logarithmsLogarithm functions, see Logarithmic

functionsLogarithmic equality, principle of, 905Logarithmic equations, 880, 906

converting to exponentialequations, 880

solving, 881, 906Logarithmic functions, 879. See also

Logarithms.graphs, 878, 898, 899inverse of, 884, 896

Logarithms, 878, 926. See alsoLogarithmic functions.

base a of a, 883, 926base a of 1, 882, 926base e, 895base 10, 883of the base to a power, 892, 926on a calculator, 883change-of-base formula, 896common, 883difference of, 890, 926and exponents, 879

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Logarithms (continued)Napierian, 895natural, 895of one, 882, 926of powers, 889, 926of products, 889, 926properties of, 889–892, 926of quotients, 890, 926sum of, 889, 926

Loudness of sound, 911

MMathematical model, 531, 825, 911Matrices, 984

row-equivalent, 987Matrix, 984. See also Matrices.Maximum value of quadratic

function, 818, 820Mean, 973Measure of central tendency, 973Median, 973, 974Midpoint formula, 993Minimum value of quadratic

function, 818, 820Missing terms, 252Mixed numerals, 949Mixture problems, 575–579Mode, 974Model, mathematical, 531, 825, 911Monomials, 245

as divisors, 295, 296 and multiplying, 267, 268

Motion formula, 448, 579Motion problems, 448, 579–581, 778Multiple, least common, 942, 943Multiples, 942

least common, 942, 943Multiplication

associative law, 56, 76commutative law, 55, 76of complex numbers, 737distributive laws, 57, 58, 76with exponential notation, 224, 227,

239of exponents, 232, 239using fraction notation, 948of functions, 989by 1, 395by �1, 66of polynomials, 267–270, 274–279,

289, 302of radical expressions, 697, 707of rational expressions, 395, 401of real numbers, 37–40, 49with scientific notation, 236, 237by zero, 38

Multiplication principlefor equations, 88, 160for inequalities, 143, 160, 618, 669

Multiplication property of zero, 38

Multiplicative identity, 53, 945Multiplicative inverse, 45, 951. See

also Reciprocals.Multiplying

using the distributive law, 58exponents, 232, 239by �1, 66by 1, 395, 949

NNapierian logarithms, 895Natural logarithms, 895Natural numbers, 10, 940Nature of solutions, quadratic

equation, 788Negative exponents, 226, 227, 239Negative integers, 11Negative numbers

on a calculator, 13square root of, 678

Negative onemultiplying by, 66property of, 66

Negative rational exponents, 691Negative square root, 678No solutions, equations with, 100Nonlinear graphs, 179Nonnegative rational numbers, 945Notation

exponential, 22, 954fraction, 945function, 484

composite, 870inverse, 862

interval, 615, 616radical, 679for rational numbers, 13, 14scientific, 235, 236, 239, 302set, 10, 142, 615

Number line, 13addition on, 22distance on, 643and graphing rational numbers, 13order on, 15, 16

Numbersarithmetic, 945complex, 736, 737composite, 941factoring, 940graphing, 13imaginary, 736, 748integers, 10irrational, 14multiples of, 942natural, 10, 940negative, 11, 13opposites, 10, 11, 24, 25, 46order of, 15, 16positive, 11prime, 940, 941

rational, 12nonnegative, 945

real, 15signs of, 26whole, 10

Numerator, 945

OOdd integers, consecutive, 125Odd root, 683, 684Offset of a pipe, 734Ohm’s law, 463One

equivalent expressions for, 946as an exponent, 223, 227, 239identity property of, 53, 76, 945logarithm of, 882, 926multiplying by, 395removing a factor of, 54, 396, 946

One-to-one function, 862, 863Operations, order of, 69, 70, 955Operations, row-equivalent, 987Opposites, 10, 11, 24, 25, 46

and changing the sign, 26, 66, 260and multiplying by �1, 66of an opposite, 25of polynomials, 260in rational expressions, 398and subtraction, 31, 260of a sum, 66 sum of, 26

Or, 634, 635Order

ascending, 251descending, 250on number line, 15, 16of operations, 69, 70, 955

Ordered pair, 166Ordered triple, 587Ordinate, 166Origin, 166Output, 484

PPairs, ordered, 166Parabola, 797

axis of symmetry, 797graphs of, 797–803, 808–811, 821intercepts, 754, 812line of symmetry, 797vertex, 797

Parallel lines, 518, 540Parentheses

in equations, 98within parentheses, 68removing, 67

Pendulum, period, 727, 779Percent, applications of, 114Perfect square, 681

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Perfect-square trinomial, 342, 343,388. See also Trinomial square.

Period of a pendulum, 727, 779Perpendicular lines, 519, 540pH, 912Pi ( ), 14Plotting points, 166Point–slope equation, 526, 527, 540Points, coordinates of, 166 Polynomial equation, 246Polynomial inequality, 832Polynomials, 245

addition of, 259, 288additive inverse, 260ascending order, 251binomials, 252coefficients, 249combining (or collecting) like (or

similar) terms, 250degree of, 251, 287descending order, 250division of, 295–298

synthetic, 976evaluating, 245, 248, 286factoring

ac-method, 336common factors, 311completely, 347difference of cubes, 352, 354, 964,

965difference of squares, 345, 354,

388, 964, 965by FOIL method, 327, 961by grouping, 313perfect-square trinomials, 343,

388, 963, 965strategy, 357sum of cubes, 352, 354, 964, 965by trial and error, 317trinomial squares, 343, 388, 963,

964trinomials, 317–322, 327–331,

336, 343, 959–963, 965and geometry, 261like terms in, 249missing terms, 252monomials, 245, 252multiplication, 267–270, 274–279,

289, 302opposite, 260perfect-square trinomial, 342prime, 321in several variables, 286subtraction, 260, 288terms, 248, 287trinomial squares, 342trinomials, 252value of, 245

Population decay, see Index ofApplications

Population growth, see Index ofApplications

Positive integers, 11Power, 954. See also Exponents.Power raised to a power, 232, 239Power rule

for exponents, 232, 239for logarithms, 889, 926

Powers on a graphing calculator, 692of i, 739logarithm of, 889, 926principle of, 717, 748

Prime factorization, 941and LCM, 942

Prime number, 940, 941Prime polynomial, 321Principal square root, 678, 681Principle

absolute-value, 644, 669addition

for equations, 83, 160for inequalities, 617, 669

of exponential equality, 904of logarithmic equality, 905multiplication

for equations, 88, 160for inequalities, 618, 669

of powers, 717, 748of square roots, 756, 840of zero products, 367, 388

Problem solving, five-step strategy,122. See also Index ofApplications.

Procedure for solving equations, 99Product. See also Multiplication.

of binomials, 268, 274–277, 302of functions, 989logarithms of, 889, 926raising to a power, 233, 239of sums and differences, 275, 302

Product rule for exponential notation, 224, 227,

239for logarithms, 889, 926for radicals, 697

Profit, 602Properties of absolute value, 642, 669Properties of exponents, 692Properties, identity

of one, 53, 76, 945of zero, 53, 76, 945

Properties of logarithms, 889–892,926

Properties of reciprocals, 46Properties of square roots, 678Property of �1, 66Proportion, 450Proportional, 450

directly, 462inversely, 464

Proportionality, constant of, 462, 464Pythagorean theorem, 379, 388, 729

QQuadrants, 167Quadratic equations, 366. See also

Quadratic functions. approximating solutions, 770discriminant, 788, 840reducible to quadratic, 790solutions, nature of, 788solving, 770

by completing the square, 762by factoring, 368graphically, 773on a graphing calculator, 371by principle of square roots, 756by quadratic formula, 768

standard form, 754, 755writing from solutions, 789

Quadratic in form, equation, 790Quadratic formula, 768, 840Quadratic functions

fitting to data, 821, 825graphs of, 797–803, 808–811, 821intercepts, 754, 812maximum value of, 818, 820minimum value of, 818, 820

Quadratic inequalities, 832Quadratic regression, 821, 825Quartic function, 821Quotient

of functions, 989of integers, 44logarithm of, 890, 926of polynomials, 295–298of radical expressions, 700, 701raising to a power, 234, 239roots of, 701, 748

Quotient rulefor exponents, 225, 227, 239for logarithms, 890, 926for radicals, 700

RRadical equations, 717Radical expressions, 679

and absolute value, 681, 682, 683,684

addition of, 706conjugates, 707, 748dividing, 700factoring, 698index, 683like, 706multiplying, 697, 707product rule, 697quotient rule, 700radicand, 679rationalizing denominators, 712,

714simplifying, 684, 693, 697, 698, 701subtraction of, 706

�

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Radical function, 680Radical symbol, 679Radicals, see Radical expressionsRadicand, 679Radius, 994Raising a power to a power, 232, 239Raising a product to a power, 233, 239Raising a quotient to a power, 234,

239Range

of a function, 482, 483, 497of a relation, 483

Rate, 450of change, 201, 509. See also Slope.exponential decay, 917exponential growth, 914

Ratio, 450Rational equations, 436Rational exponents, 690, 691Rational expressions, 394. See also

Fraction expressions.addition, 414–417complex, 430dividing, 405multiplying, 395, 399reciprocal of, 405simplifying, 396subtraction, 422–425undefined, 394

Rational inequalities, 836Rational numbers, 12

decimal notation, 13, 14, 15as exponents, 690, 691nonnegative, 945

Rationalizing denominators, 712, 714Real numbers, 15

addition, 22, 23, 49division, 44–49multiplication, 37–40, 49order, 15subsets, 10, 15subtraction, 30, 31, 49

Real-number system, 15. See also Realnumbers.

Reciprocals, 45, 951and division, 47, 405, 952and exponential notation, 691properties, 46of rational expressions, 405sign of, 46

Rectanglearea, 3Golden, 787

Reducible to quadratic equations, 790Reflection, 867Regression on a graphing calculator,

534, 825, 918Related equation, 655Relation, 483, 497, 860

inverse, 860, 861Remainder, 297, 298

Removing a factor of 1, 54, 396, 946Removing parentheses, 67Repeating decimals, 14Revenue, 602Reversing equations, 90Review of introductory algebra, 968,

969Right circular cylinder, surface area,

291Right triangle, 378Rise, 198, 505, 506Road-pavement messages, 730Roots

approximating, 685cube, 682even, 684odd, 683, 684square, 678

Roster notation, 10Row-equivalent matrices, 987Row-equivalent operations, 987Rows of a matrix, 984Run, 198, 505, 506

SScientific notation, 234, 235, 239, 302

on a calculator, 237converting from/to decimal

notation, 235, 236dividing with, 237multiplying with, 236, 237

Second coordinate, 166Second-order determinant, 979Semiannually compounded interest,

853, 926Set-builder notation, 142, 615Sets, 10

disjoint, 633empty, 633intersection, 630, 669 notation, 10, 42, 615roster notation, 10set-builder notation, 142solution, 139subset, 10union, 634, 635, 669

Several variables, polynomial in, 286Sighting to the horizon, 726Sign changes in fraction notation, 48Sign of a reciprocal, 46Signs of numbers, 26Similar triangles, 452, 453Simple interest, 131Simplest fraction notation, 946Simplifying

checking, 400complex rational (or fraction)

expressions, 430–433fraction expressions, 396, 946

fraction notation, 946radical expressions, 684, 693, 697,

698, 701rational expressions, 396removing parentheses, 67

Skidding car, speed, 727Slope, 198, 210, 505, 508, 540

applications, 200–202from equations, 203graphing using, 515of a horizontal line, 204, 210, 517not defined, 517of parallel lines, 518, 540of perpendicular lines, 519, 540as rate of change, 201, 509of a vertical line, 204, 210, 517zero, 517

Slope–intercept equation, 508, 540Solution sets, 139, 614Solutions

of equations, 82, 168, 175, 742extraneous, 718of inequalities, 139, 614, 653nature of, quadratic equations, 788of system of equations, 549of systems of inequalities, 657writing equations from, 789

Solve, in problem-solving process,122

Solving equations, 99with absolute value, 644, 645, 647using addition principle, 83, 160clearing decimals, 96, 98clearing fractions, 96, 97collecting like terms, 95containing parentheses, 98exponential, 904by factoring, 368fraction, 436graphically, 553, 719, 773on a graphing calculator, 371logarithmic, 881, 906using multiplication principle, 88,

160containing parentheses, 98using principle of zero products,

368procedure, 99quadratic, 770

by completing the square, 762by factoring, 368graphically, 773on a graphing calculator, 371by principle of square roots, 756by quadratic formula, 768

quadratic in form, 790radical, 717, 719, 721rational, 436reducible to quadratic, 790systems, see Systems of equations

Solving formulas, 107–109, 781

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Solving inequalities, 139, 614, 617,618, 632, 635

with absolute value, 646, 647using addition principle, 141, 160using multiplication principle, 143,

160polynomial, 832, 833, 835rational, 836, 837solution set, 139systems of, 657

Solving systems of equations, seeSystems of equations

Soundloudness, 911speed, 723

Special products of polynomialssquaring binomials, 276, 277, 302sum and difference of two

expressions, 275, 302two binomials (FOIL), 274, 302

Speed, 450of a skidding car, 727of sound, 723

Square, completing, 759Square of a binomial, 276, 277, 302Square roots, 678

approximating, 14, 685negative, 678of negative numbers, 678principal, 678, 681principle of, 756, 840properties, 678

Square viewing window, 520Square-root function, 680Squares

of binomials, 276, 277, 302differences of, 344, 354, 964, 965perfect, 681sum of, 347, 354, 965trinomial, 342

Squaring a number, 678Squaring the window, 520Standard form

of circle equations, 944of quadratic equations, 754, 755

Standard viewing window, 180Stat feature on a graphing calculator,

535State the answer, 122Subset, 10Substituting, 3Substitution method, 557Subtraction, 30

by adding an opposite, 31, 260of complex numbers, 737of exponents, 225, 227, 239using fraction notation, 950of functions, 989of logarithms, 890, 926opposites and, 31, 260of polynomials, 260, 288

of radical expressions, 706of rational expressions, 422–425of real numbers, 30, 31, 49

Sumof cubes, factoring, 352, 354, 964,

965of functions, 989of logarithms, 889, 926opposite of, 66of opposites, 26of squares, 347, 354, 965

Sum and difference, product of, 275,302

Supplementary angles, 562Supply and demand, 604, 629Surface area

cube, 110 right circular cylinder, 291

Symmetry, line of, 797, 811Synthetic division, 976Systems of equations, 548

consistent, 551, 552, 566with dependent equations, 552,

566, 982graphs of, 549–553inconsistent, 551, 552, 566, 982with independent equations, 552,

566with infinitely many solutions, 552,

566with no solution, 551, 552, 566solution of, 549, 587solving

comparing methods, 568Cramer’s rule, 979–982elimination method, 563, 567, 588graphically, 549, 550using matrices, 984substitution method, 557

Systems of inequalities, 657

TTable feature on a graphing

calculator, 101, 175, 485Table of primes, 941Terminating decimals, 14Terms, 58

coefficients of, 249collecting (or combining) like, 61,

250, 287degree of, 251, 287of an expression, 58like, 61, 249, 287missing, 252of a polynomial, 248, 287

Theorem, Pythagorean, 379, 388, 729 Third-order determinant, 979Total cost, 602Total profit, 602Total revenue, 602

Total-value problems, 573Trace feature on a graphing

calculator, 485Translating

to algebraic expressions, 5to equations, 122to inequalities, 151in problem solving, 122

Translating for Success, 133, 381, 454,582, 623, 731, 782, 920

Translations of graphs, 503, 800, 849,850

Travel of a pipe, 734Tree, factor, 941 Trial-and-error factoring, 317Triangle

right, 378sum of angle measures, 129

Triangles, similar, 452, 453Trinomial square, 342

factoring, 343, 388, 963, 965Trinomials, 252

factoring, 317–322, 327–331, 336,343, 388, 959–963, 965

perfect-square, 342, 963, 965Triple, ordered, 587True equation, 82True inequality, 139, 614

UUndefined rational expression, 394Undefined slope, 204, 517Union, 634, 635, 669Units, changing, 971

VValue

of an expression, 3of a function, 484, 485of a polynomial, 245

Value feature on a graphingcalculator, 485

Variable, 2substituting for, 3

Variable cost, 602Variation

constant, 462, 464direct, 462inverse, 464joint, 467

Variation constant, 462, 464Vars on a graphing calculator, 535Vertex of a parabola, 797Vertical line, 189, 210, 517, 540

slope, 204, 210Vertical-line test, 487Vertices of a system of inequalities,

660

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Viewing window, 180, 188square, 520

Visualizing for Success, 191, 281, 521,663, 813, 900

WWalking speed, 921Whole numbers, 10, 945Wind chill temperature, 735Window, viewing, 180, 188

squaring, 520

Work principle, 448Work problems, 446

Xx-intercept, 186, 513, 754, 812

Yy-intercept, 175, 186, 210, 504, 513,

812, 849

ZZero

degree of, 251dividend of, 45division by, 44, 45as an exponent, 223, 227, 239identity property of, 23, 53, 76, 945multiplication property, 38slope, 204, 517

Zero feature on a graphing calculator,371

Zero products, principle of, 367, 388

I-10

INDEX

Co

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200

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Geometric Formulas

PLANE GEOMETRY

Rectangle

Area:Perimeter:

Square

Area:Perimeter:

Triangle

Area:

Sum of Angle Measures

Right Triangle

Pythagorean Theorem:

Parallelogram

Area:

Trapezoid

Area:

Circle

Area:Circumference:

and 3.14 are differentapproximations for ��22

7

C � � � d � 2 � � � r

A � � � r 2

A � 12 � h � �a � b�

A � b � h

a2 � b2 � c2

A � B � C � 180�

A � 12 � b � h

P � 4 � sA � s2

P � 2 � l � 2 � wA � l � w

SOLID GEOMETRY

Rectangular Solid

Volume:

Cube

Volume:

Right Circular Cylinder

Volume:Surface Area:

Right Circular Cone

Volume:Surface Area:

Sphere

Volume:Surface Area: S � 4 � � � r 2

V � 43 � � � r 3

S � � � r 2 � � � r � sV � 1

3 � � � r 2 � h

S � 2 � � � r � h � 2 � � � r 2

V � � � r 2 � h

V � s3

V � l � w � hw

l

s

s

h

b

A

B

C

a

b

c

h

b

h

b

a

r

d

l w

h

ss

s

r

r

h

hs

r

r

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Mat116 Appendices Student Solutions

Documents

handling dimension

a1 b1 c1

handling dimension

treat dimension

a2 a3 a3

3z 2z 4z

4x 8x 2x

2y 2y 4y