Chapter 1 Expression, Equations, and Inequalities

Unit Essential Questions:• How do variables help you

model real-world situations?• How can you use properties

of real numbers to simplify algebraic expressions?

• How do you solve an equation or inequality?

Chapter 1Expression, Equations, and Inequalities

Students will be able to identify and describe patterns

Section 1.1Patterns and Expressions

1) 6 + (-6) 2) 62

5+ 4

3

10

3) 61 - (-11) 4) 5

2-

13

4

Warm UpEvaluate.

0 107/10

72 -3/4

Key Concepts

Variable - a symbol, usually a letter, that represents one or more numbers

Numerical Expression - a mathematical phrase that contains numbers and operation symbols

Algebraic Expression - a mathematical phrase that contains one or more variables

Example 1Describe each pattern using words. Draw the next figure in the pattern.

a)

b)

The bottom row increases by one

One square is added to the right bottom

Example 2These figures are made with toothpicks.

a) How many toothpicks are in the 20th figure? Use a table of values with a process column to justify your answer.

a) What expression describes the number of toothpicks in the nth figure?

Figure Process Output

1 1 x 5 5

2 2 x 5 10

3 3 x 5 15

… … …

20 20 x 5 100

The 20th figure has 100 toothpicks.

5n

Example 3Identify a pattern by making a table of the inputs and

outputs. Include a process column.

a) b)

Input Process Output

1 1 x 1 1

2 2 x 1 2

3 3 x 1 3

4 4 x 1 4

5 5 x 1 5

Input Process Output

1 6 - 1 5

2 6 - 2 4

3 6 - 3 3

4 6 - 4 2

5 6 - 5 1

Example 4Identify a pattern and find the next three numbers in

the pattern.

a) 2, 4, 8, 16, … multiply by 2; 32, 64, 128

b) 4, 8, 12, 16, …add 4; 20, 24, 28

c) 5, 25, 125, 625, …multiply by 5; 3125, 15625, 78125

Students will be able to graph and order real numbers.

Students will be able to identify properties of real numbers.

Section 1.2Properties of Real Numbers

Warm UpWrite each number as a percent.

1) 0.5 2) 0.25 3) 1

3

4) 12

5 5) 1.72 6) 1.23

50% 25% 33.3%

140% 172% 123%

-15, -7, -4, 0, 4, 7{…, -2, -1, 0, 1, 2, 3, …}

Add the negative natural numbers to the whole numbers

Integers

Z

0, 4, 7, 15{0, 1, 2, 3, … }

Add 0 to the natural numbers

Whole Numbers

W

4, 7, 15{1, 2, 3, …}

These are the counting numbers

Natural Numbers

N

ExamplesDescriptionName

Key ConceptsSubsets of the Real Numbers

This is the set of numbers whose decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers.

Irrational NumbersI

These numbers can be expressed as an integer divided by a nonzero integer:Rational numbers can be expressed as terminating or repeating decimals.

Rational NumbersQ

ExamplesDescriptionName

Key ConceptsSubsets of the Real Numbers

2 » 1.414214

- 3 » -1.73205

p » 3.142

-p

2» -1.571

-17 =-17

1,-5 =

-5

1,-3, -2

0,2,3,5,17

2

5= 0.4,

-2

3= -0.666666... = -0.6

Rational Numbers

The Real Numbers

Irrational Numbers

Integers

Whole Numbers

Natural Numbers

The set of real numbers is formed by combining the rational numbers and the irrational numbers.

Example 1Classify and graph each number on a number line.

a) 3

b) 11

5

c) - 11

natural, whole, integer, rational

rational

irrational

Example 2Compare the two numbers. Use < and >.

a) -5, -8

-5 > -8

b) 1/3, 1.333

1/3 < 1.333

c)3, √3

3 > √3

Key ConceptsLet a, b, and c be real numbers.

Opposite - (additive inverse) the opposite of any number ais -a.

Reciprocal - (multiplicative inverse) the reciprocal of any nonzero number a is 1/a.

Property Addition Multiplication

Commutative

Associative

Identity

Inverse

Distributive

(ab) ×c = a ×(bc) (a + b) +c = a + (b +c)

a + b = b +a ab = ba

a + 0 = a a ×1 = a

a + (-a) = 0 a ×

1

a= 1

a(b +c) = ab +ac

Example 3Name the property of real numbers illustrated by each

equation.

a) n · 1 = n

Multiplicative Identity

b) a (b + c) = ab + ac

Distributive Property

c) 4 + 8 = 8 + 4

Commutative Property of Addition

d) 0 = q + (-q)

Additive Inverse

Students will be able to evaluate algebraic expressions

Students will be able to simplify algebraic expressions

Section 1.3Algebraic Expressions

Warm Up

Use order of operations to simplify.

1) 3 ¸ 4 + 6 ¸ 4 2) 5[(2+ 5) ¸ 3]

3) 8+5 ´ 2

12 4) 40+24 ¸ 8 - 22 - 1

9/4 35/3

3/2 38

Key Concepts

Term - an expression that is a number, a variable, or the product of a number and one or more variables

Coefficient - the numerical factor of a term

Constant Term - a term with no variable

Like Terms - the same variables raised to the same power

Example 1Write an algebraic expression that models each

word phrase.

a) six less than a number w

w - 6

a) the product of 11 and the difference of 4

and a number r

11(4 – r)

Example 2Evaluate each expression for the given values of

the variables.

a) 6c + 5d - 4c - 3d + 3c - 6d; c = 4 and d = -2

5c – 4d = 5(4) – 4(-2) = 20 + 8 = 28

b) 10a + 3b - 5a + 4b + 1a + 5b; a = -3 and b = 5

6a + 12b = 6(-3) + 12(5) = -18 + 60 = 42

Example 3Simplify by combining like terms

a) 4 + 3t - 2t b) 3 - 2(2r - 4)

c) 9y + 2x - 4y + x d) -(j - 3j + 8)

t + 4 -4r + 113 – 4r + 8

3x + 5y -j + 3j - 8

2j - 8

Example 4

Write an algebraic expression to model the situation.

You fill your car with gasoline at a service station for $2.75 per gallon. You pay with a $50 bill. How much change will you receive if you buy g gallons of gasoline? How much change will you receive if you buy 14 gallons?

$50 – 2.75g

$50 – $2.75(14) = $11.50

Students will be able to solve equations

Students will be able to solve problems by writing equations

Section 1.4Solving Equations

Warm Up

Simplify.

1) 4x +3x-4 2) -p

3+

q

3-

2p

3- q

3) - 2(4 + b) + 4(b - 5) 4) (k - m) - (m - k)

7x - 4

-p -

2q

3

2b – 28 2k – 2m

Key ConceptsProperties of Equality

Let a, b, and c represent real numbers

Property DefinitionReflexive a = a

Symmetric If a = b, then b = a

Transitive If a = b and b = c, then a = c

Substitution If a = b, then you can replace a with b and vise versa

Addition/ Subtraction

If a = b, then a + c = b + c

and a - c = b - c

Multiplication/ Division

If a = b and c = 0, then

ac = bc and a/c = b/c

Example 1Solve each equation. Check your answers.a) 18 - n = 10 b) 3.5y =14

c) 5 - w = 2w -1 d) -2s = 3s - 0

n = 8

s = 0w = 2

y = 4

Example 2Solve each equation. Check your answers.a) 2(x + 3) + 2(x + 4) = 24 b) 8z + 12 = 5z - 21

c) 7b - 6(11 - 2b) = 10 d) 10k - 7 = 2(13 - 5k)

2x + 6 + 2x + 8 = 24

4x + 14 = 24

4x = 10

x = 5/2

3z + 12 = -21

3z = -33

z = -11

7b - 66 + 12b = 10

19b - 66 = 10

19b = 76

b = 4

10k - 7 = 26 – 10k

20k - 7 = 26

20k = 33

k = 33/20

Key Concepts

Identity - an equation that is true for every value of the variable.

Literal Equation - an equation that uses at least 2 letters as variables. You can solve for any variable “in terms of” the other variables.

Example 3Determine whether the equation is sometimes,

always, or never true.

a) 3x - 5 = -2 b) 2x - 3 = 5 + 2x

c) 6x -3(2 + 2x) = -6

3x = 3

x = 1

Sometimes

-3 = 5

Never

Always

6x – 6 - 6x = -6

-6 = -6

Example 4Solve each formula for the indicated variable.

a) ax + bx -3 = -4, for x b) V =

1

3pr2h, for h

ax + bx = -1

x(a + b) = -1

x =

-1

a + b

3V = pr2h

h =

3V

pr2

Students will be able to solve and graph inequalities

Section 1.5 Part 1Solving Inequalities

Warm Up

State whether the inequality is true or false.

1) 5 < 12 2) 5 < -12 3) 5 ≥ 5

True False True

Key Concepts

Writing and graphing inequalities

x is greater than 4

x is greater than or equal to 4

x is less than 4

x is less than or equal to 4

x > 4

x ³ 4

x < 4

x £ 4

4

4

4

4

Example 1

Write an inequality that represents the sentence.

a) The product of 12 and a number is less than 6.

12x < 6

b) The sum of a number and 2 is no less than the

product of 9 and the same number.

x + 2 ≥ 9x

Example 2Solve each inequality. Graph the solution.

a) 3x - 8 > 1 b) 3v ≤ 5v + 18

c) 7 – x ≥ 24 d) 2(y - 3) + 7 < 21

3x > 9

x > 3

-2v ≤ 18

v ≥ -9

- x ≥ 17

x ≤ -17

2y – 6 + 7 < 21

2y + 1 < 212y < 20

y < 10

3 -9

-17

10

Example 3

Is the inequality always, sometimes, or never true?

a) - 2(3x + 1) > - 6x + 7 b) 5(2x - 3) - 7x ≤ 3x + 8

c) 6(2x – 1) ≥ 3x + 12

Never

-6x - 2 > -6x + 7- 2 > 7

10x – 15 – 7x ≤ 3x + 83x – 15 ≤ 3x + 8

– 15 ≤ 8

Always

12x – 6 ≥ 3x + 129x – 6 ≥ 12

x ≥ 29x ≥ 18 Sometimes

Students will be able to write and solve compound inequalities

Section 1.5 Part 2Solving Inequalities

Warm UpYou want to download some new songs on your

MP3 player. Each song will use about 4.3 MB of space. You have 7.8 GB of 19.5 GB available on our MP3 player. At most, how many songs can you download? (1 GB = 1024 MB)

7.8GB = 7987.2 MB

4.3x ≤ 7987.2

x ≤ 1857.49

You can download 1857 songs

Key Concepts

Compound Inequalities - two inequalities joined with the word and or the word or.

AND means that a solution makes BOTHinequalities true.

OR means that a solution makes EITHERinequality true.

Example 1Solve each compound inequality. Graph the solution.

a) 4r > -12 and 2r < 10 b) 5z ≥ -10 and 3z < 3

r > -3 r < 5and z ≥ -2 z < 1and

Example 2Solve each compound inequality. Graph the

solution.

a) -2 < x + 1 < 4 b) 3 < 5x - 2 < 13-3 < r < 3 5 < 5x < 15

1 < x < 3

Example 3Solve each compound inequality. Graph the

solution.

a) 3x < -6 or 7x > 35 b) 5p ≥ 10 or -2p > 10x < -2 x > 5or p ≥ 2 p < -5or

Students will be able to write and solve equations involving absolute value

Section 1.6 Part 1Absolute Value Equations and Inequalities

Warm Up

Solve each equation.

1) 6x - 6(10 - x) = 15 2) 12x - 4 = 2(11 + x)6x – 60 + 6x = 15

12x - 60 = 15

12x = 75

x = 25/4

12x – 4 = 22 + 2x

10x - 4 = 22

10x = 26

x = 13/5

Key Concepts

Absolute Value - the distance from zero on the number line. Written |x|

Extraneous Solution - a solution derived from an original equation that is NOT a solution to the original equation.

Key Concepts

Steps to solve an absolute value equation:

1) Isolate the absolute value expression

2) Write as two equations (set expression in the absolute value to the positive and negative - absolute value sign goes away)

3) Solve for each equation

4) Check for extraneous solutions

Example 1Solve. Check your answers.

2x - 1 = 5

2x – 1 = 5

2x = 6

x = 3

2x – 1 = -5

2x = -4

x = -2

Check:|2(3) – 1| = 5|2(-2) – 1| = 5|5| = 5 |-5| = 5

Example 2Solve. Check your answers.

3 x + 2 - 1 = 8

x + 2 = 3

x = 1

x + 2 = -3

x = -5

Check:3|(1) + 2|- 1 = 8 3|(-5) + 2|- 1 = 83|3|-1 = 8 3|-3| - 1 = 89 – 1 = 8 9 – 1 = 8

3|x + 2| = 9

|x + 2| = 3

Example 3Solve. Check your answers.

3x + 2 = 4x + 5

3x + 2 = 4x + 5

- x = 3

x = -3

3x + 2 = -4x – 5

7x = -7

x = -1

Check:|3(-3) + 2| = 4(-3) + 5 |3(-1) + 2| = 4(-1) + 5|-7| = -7 |-1| = 1

Students will be able to write and solve inequalities involving absolute value

Section 1.6 Part 2Absolute Value Equations and Inequalities

Warm UpYou are riding an elevator and decide to find out

how far it travels in 10 minutes. You start at the third floor and record each trip. If each floor is 12ft, how far did the elevator travel?

Trip 1 2 3 4 5

Floors +8 -6 +9 -3 +7

8 + |-6| + 9 + |-3| + 7 = 33

33(12 ft) = 396ft

Key ConceptsSteps to solve an absolute value inequality:

1) Isolate the absolute value expression

2) Write as a compound inequality

3) Solve the inequalities

A < b or A £ b write compound inequality as AND

A > b or A ³ b write compound inequality as OR

Example 1Solve the inequality. Graph the solution.

2x - 1 < 5

2x – 1 < 5

2x < 6

x < 3

2x – 1 > -5

2x > -4

x > -2and

Example 2Solve the inequality. Graph the solution.

22y - 5 + 6 ³ 16

|2y – 5| ≥ 5

2x - 5 ≥ 5

x ≥ 5

2x ≥ 102x – 5 ≤ -5

x ≤ 0 or

2|2y – 5| ≥ 10

2x ≤ 0