Algebra 2 Chapter 1 Notes Equations and Inequalities

Algebra 2

Chapter 1 Notes

Equations and Inequalities

1

Algebra is the ability to move values from one side of the

equation or inequality to the other side by doing the opposite

operation.

x + 2

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Common Assumptions with Numbers

+ 1 n. 1

1

• The sign of a number is positive, +• The coefficient is 1• The decimal point is to the right of the number• As a whole number it is over 1• The power of the number is 1

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Origin

-3 -2 -1 0 1 2 3

Real Numbers and Number Operations

Whole numbers = 0, 1, 2, 3 …

Integers = …, -3, -2, -1, 0, 1, 2, 3 …

Rational numbers = numbers such as 3/4 , 1/3, -4/1 that can be written as a ratio of the two integers. When written as decimals, rational numbers terminate or repeat, 3/4 = 0.75, 1/3 = 0.333…

Irrational numbers = real numbers that are NOT rational, such as, and π , When written as decimals, irrational numbers neither terminate or repeat.

A Graph of a number is a point on a number line that corresponds to a real number

The number that corresponds to a point on a number line is the Coordinate of the point.

1.1

4

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

Graph - 4/3, 2.7, 2

Graph - 2, 3

Graph - 1, - 3

• •

• •

••

•

1.1Real Numbers and Number Operations

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- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

Example: You can use a number line to graph and order real numbers.

Increasing order (left to right): - 4, - 1, 0.3, 2.7

Properties of real numbers include the closure, commutative, associative, identity, inverse and distributive properties.

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Real Numbers and Order of Operation 1.1

Using Properties of Real Numbers 1.1

Properties of addition and multiplication [let a, b, c = real numbers]

Property Addition Multiplication

Closure a + b is a real number a • b is a real number

Commutative a + b = b + a a • b = b • a

Associative ( a + b ) + c = a + ( b + c ) ( a b ) c = a ( b c )

Identity a + 0 = a , 0 + a = a a • 1 = a , 1 • a = a

Inverse a + ( -a ) = 0 a • 1/a = 1 , a 0

Distributive a ( b + c) = a b + a c

Opposite = additive inverse, for example a and - a

Reciprocal = multiplicative inverse (of any non-zero #) for example a and 1/a

Definition of subtraction: a – b = a + ( - b )

Definition of division: a / b = a 1 / b , b 0

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Identifying properties of real numbers & number operations

( 3 + 9 ) + 8 = 3 + ( 9 + 8 ) 14 • 1 = 14

[ Associative property of addition ] [Identity property of multiplication ]

Operations with real numbers:

Difference of 7 and – 10 ? 7 – ( - 10 ) = 7 + 10 = 17

- 10 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 7

••

Quotient of - 24 and 1/3 ?

Real Numbers and Number Operations 1.1

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Real Numbers and Number Operations 1.1

Give the answer with the appropriate unit of measure

A.) 345 miles – 187 miles = 158 miles

B.) ( 1.5 hours ) ( 50 miles ) = 75 miles 1 hour

C) 24 dollars = 8 dollars per hour 3 hours

D) ( 88 feet ) ( 3600 seconds ) ( 1 mile ) = 60 miles per hour 1 second 1 hour 5280 feet

“Per” means divided by

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Algebraic Expressions and Models 1.2

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Order of Operations1. First, do operations that occur within grouping symbols - 4 + 2 ( -2 + 5 ) 2 = - 4 + 2 (3 ) 2

2. Next, evaluate powers = - 4 + 2 ( 9 )3. Do multiplications and divisions from left to right = - 4 + 184. Do additions and subtractions from left to right = 14

Numerical expression: 25 = 2 • 2 • 2 • 2 • 2[ 5 factors of 2 ] or [ 2 multiplied out 5 times ]

In this expression:the number 2 is the basethe number 5 is the exponentthe expression is a power.

A variable is a letter used to represent one or more numbers. Any number used to replace variable is a value of the variable. An expression involving variables is called an algebraic expression. The value of the expression is the result when you evaluate the expression by replacing the variables with numbers.

An expression that represents a real-life situation is a mathematical model. See page 12.

Algebraic Expressions and Models 1.2

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Example: You can use order of operations to evaluate expressions.

Numerical expressions: 8 (3 + 42) – 12 ÷ 2 = 8 (3 + 16) – 6 = 8 (19) – 6 = 152 – 6 = 146

Algebraic expression: 3 x2 – 1 when x = – 5

3 (– 5 )2 – 1 =

3 (25) – 1 = 74

Sometimes you can use the distributive property to simplify an expression.

Combine like terms: 2 x2 – 4 x + 10 x – 1 =

2 x2 + (– 4 + 10 ) x – 1 =

2 x2 + 6 x - 1

Evaluating Powers 1.2

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Example 1: ( - 3 ) 4 = ( - 3 ) ( - 3 ) ( - 3 ) ( - 3 ) = 81

- 3 4 = - ( 3 3 3 3 ) = - 81

Example 2: Evaluating an algebraic expression

- 3 x 2 – 5 x + 7 when x = - 2

- 3 ( - 2 ) 2 – 5 ( - 2 )x + 7 [ substitute – 2 for x ]

- 3 ( 4 ) – 5 ( - 2 )x + 7 [ evaluate the power, 2 2 ]

- 12 + 10 + 7 [ multiply ]

+ 5 [ add ]

Example 3: Simplifying by combining like terms

a) 7 x + 4 x = ( 7 + 4 ) x [ distributive ]= 11 x [ add coefficients ]

b) 3 n 2 + n – n 2 = ( 3 n 2 – n 2 ) + n [ group like terms ] = 2 n 2 + n [ combine like terms ]

c) 2 ( x + 1 ) – 3 ( x – 4 ) = 2 x + 2 – 3 x + 12 [ distributive ] = ( 2 x – 3 x ) + ( 2 + 12 ) [ group like terms ] = - x + 14 [ combine like terms ]

Solving Linear Equations 1.3

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Transformations that produce equivalent equations

Additional property of equality Add same number to both sidesif a = b, then a + c = b + c

Subtraction property of equality Subtract same number to both sidesif a = b, then a - c = b - c

Multiplication property of equality Multiply both sides by the same number if a = b and c ǂ 0, then a • c = b • c

Division property of equality Divide both sides by the same number if a = b and c ǂ 0, then a ÷ c = b ÷ c

Linear Equations in one variable in form a x = b, where a & b are constants and a ǂ 0.A number is a solution of an equation if the expression is true when the number is substituted.Two equations are equivalent if they have the same solution.

Solve Linear Equations 1.3

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Solving for variable on one side [by isolating the variable on one side of equation ]

Example 1: 3 x + 9 = 15 7 3 x + 9 - 9 = 15 - 9 7 [ subtract 9 from both sides to eliminate the other term ] 3 x = 6 7

7 • 3 x = 7 • 63 7 [ multiply both sides by 7/3, the reciprocal of 3/7, to get x by

itself]x = 14

Example 2: 5 n + 11 = 7 n – 9 - 5 n - 5 n [ subtract 5 n from both sides to get the variable on one side ]

11 = 2 n – 9 + 9 + 9 [ add 9 to both sides to get rid of the other term with the

variable ]

20 = 2 n 2 2 [ divide both sides by 2 to get the variable n by itself on one

side ]

10 = n

Solve Linear Equations 1.3

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Example: You can use properties of real numbers and transformations that produce equivalent equations to solve linear equations.

Solve: – 2 ( x – 4 ) = 12

– 2 x + 8 = 12

– 2 x = 4

x = – 2

Solve Linear Equations 1.3

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Equations with fractions

Example 3: 1 x + 1 = - 1 3 4 6

Solve Linear Equations 1.3

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Identifying Properties

33. − 8 + 8 = 0

34. ( 3 • 5 ) • 10 = 3 • ( 5 • 10 )

35. 7 • 9 = 9 • 7

36. ( 9 + 2 ) + 4 = 9 + ( 2 + 4 )

37. 12 (1) = 12

38. 2 ( 5 + 11 ) = 2 • 5 + 2 • 11

Solve Word Problems 1.3

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Operations

43. What is the sum of 32 and – 7 ?

44. What is the sum of – 9 and – 6 ?

45. What is the difference of – 5 and 8 ?

46. What is the difference of – 1 and – 10 ?

47. What is the product of 9 and – 4 ?

48. What is the product of – 7 and – 3 ?

49. What is the quotient of – 5 and – ½ ?

50. What is the quotient of – 14 and 7/4 ?

Solve Unit Measures 1.3

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Unit Analysis

51. 8 1/6 feet + 4 5/6 feet =

52. 27 ½ liters – 18 5/8 liters =

53. 8.75 yards ( $ 70 ) = 1 yard

54. ( 50 feet ) ( 1 mile ) ( 3600 seconds ) = 1 second 5280 feet 1 hour

Example: You can an equation that has more than one variable, such as a formula, for one of its variables.

Solve the equation for y:

2 x – 3 y = 6

– 3 y = – 2 x + 6

y = 2 x – 2 3

Solve for the formula for the area of a trapezoid for h:

A = 1 ( b1 + b2) h 2

2 A = ( b1 + b2) h

2 A = h( b1 + b2)

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ReWriting Equations and Formulas 1.4

ReWriting an Equation with more than 1 variable 1.4

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Solve : 7 x – 3 y = 8 for the variable y.

7 x – 3 y = 8 - 7 x - 7 x [ subtract 7 x from both sides to get rid of the other term ]

– 3 y = 8 – 7 x – 3 – 3 – 3 [divide both sides by – 3 to get the variable x by itself on one side ]

y = – 8 + 7 x 3 3

Calculating the value of a variable

Solve: x + x y = 1 when x = – 1 and x = 3

x + x y = 1 [ first solve for y so that when you replace x with – 1 and 3, you also solve for y ] - x - x [ subtract x from both sides to get rid of the other term without y in it ]

x y = 1 – x x x [divide by x to get y by itself ]

y = 1 – x when x = – 1, then y = – 2 and when x = 3, then y = – 2/3 x

Common Formulas 1.4

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Distance D = r t d = distance, r = rate, t = time

Simple interest I = p r t I = interest, p = principal, r = rate, t = time

Temperature F = 9/5 C + 32 F = degrees Fahrenheit, C = degrees Celsius

Area of a Triangle A = ½ b h A = area, b = base, h = height

Area of a Rectangle A = l w A = area, l = length, w = width

Perimeter of Rectangle P = 2 l + 2 w P = perimeter, l = length, w = width

Area of Trapezoid A = ½ ( b1 + b2 ) h A = area, b1 = 1 base, b2 = 2 base, h = height

Area of Circle A = π r2 A = area, r = radius

Circumference of Circle C = 2 π r C = circumference, r = radius

ReWriting a Common Formula 1.4

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P = 2 l + 2 w [ solve for w ]– 2 l – 2 l [ subtract 2 l from both side to get rid of the other term ]

P – 2 1 = 2 w [ divide each side by 2 to get w by itself on one side of the equation ] 2 2

P – 2 l = w 2

Example 6: Applying a common formula

Problem: You have 40 feet of fencing with which to enclose a rectangular garden. Express the garden’s area in terms of its length only.

A= l w

A = l ( P – 2 l ) [ substitute P – 2 l for w ] 2 2A = l (40 – 2 l ) [ substitute 40 for P as indicated in problem above ] 2A = l ( 20 – l )

Problem Solving Using Algebraic Models 1.5

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Verbal model is a word equationwhile an Algebraic model is a math statement

Example 1, Writing and using a formula

Verbal : Distance = Rate • Time

Labels: 550 kilos = r kilos/hr • 2.25 hours

Algebraic: 550 = r • 2.25

550 = r • 2.25 2.25 2.25 [ divide by 2.25 to solve for r ]

244 = r

Key steps in problem solving plan

Write a verbal model

Assign labels

Write an algebraic model

Solve algebraic model

Answer question

Problem Solving Using Algebraic Models 1.5

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Example: create a problem solving plan in which you write a verbal model, assign labels, write and solve an algebraic model, and then answer the question.

How far can you drive at 55 miles per hour for 4 hours?

Verbal Model --- Distance = Rate Time

Labels --- Distance = d (miles), Rate = 55 (miles per hour), Time = 4 (hours)

Algebraic Model --- d = 55 ● 4

You can drive 220 miles

Problem Solving Using Algebraic Models 1.5

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Example 3, Highway Local

Verbal : Total miles = fuel efficiency x amount of gas + fuel efficiency x amount of gas

Labels: Total miles = 600 [miles] Fuel efficiency (highway) = 21 [miles per gallon] Amount of gas (highway) = x [gallons] Fuel efficiency (local) = 13 [miles per gallon] Amount of gas (local) = 40 – x [gallons]

Algebraic 600 = 21 x + 13 (40 – x)

600 = 21 x + 520 – 13 x

600 = 8 x + 520 - 520 - 520

80 = 8 x 8 8

10 = x

10 gallons x cost of gas per gallon = total cost for gas for this trip

Solving Linear Inequalities 1.6

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Transformations that produce equivalent inequalities

Add same number to both sides

Subtract same number to both sides

Multiply both sides by the same positive number

Divide both sides by the same positive number

Multiply both sides by the same negative number, then reverse the inequality

Divide both sides by the same negative number, then reverse the inequality

Linear inequalities are indicated by > , < , ≥ , ≤

Solution of an inequality in one variable is a value which makes the inequality true.

Graph of inequality consists of all points on a real # line that corresponds to solutions of the inequality.

Compound inequality is two simple inequalities joined by “and” or “or”- 2 1

- 1 2

Example 1: 5 y – 8 < 12 + 8 + 8

5 y < 205 5

y < 4

Example 2: 2 x + 1 ≤ 6 x - 1

- 6 x - 6 x

- 4 x + 1 ≤ - 1 - 1 - 1

- 4 x ≤ - 2 -4 - 4

x ½

0 4

0 ½ 1

Divide both sides by the same negative number, then reverse the inequality

≤

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Solving an Inequality with a Variable on 1 side 1.6

Example 3: - 2 ≤ 3 t – 8 ≤ 10

+ 8 + 8 + 8

6 ≤ 3 t ≤ 18 3 3 3

2 ≤ t ≤ 6

Example 4: 2 x + 3 < 5 or 4 x – 7 > 9

- 3 - 3 + 7 + 7

2 x < 2 4 x > 16 2 2 4 4

x < 1 x > 4

2 6

0 1 4

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Solving Linear Inequalities 1.6

Example: You can use transformations to solve inequalities . Reverse the inequality when you multiply or divide both sides by a negative number.

4 x + 1 < 7 x – 5

– 3 x < – 6

x > 2

0 ≤ 6 – 2 n ≤ 10

– 6 ≤ – 2 n ≤ 4

3 ≤ n ≤ – 2

2 – 2 3

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Solving Linear Inequalties 1.6

Absolute Value of a number x , | x | , is the distance the number is from 0 on a number line.

Example: | - 3 | = 3 | 3 | = 3

–3 0 + 3

3 units = 3 units

The absolute value equation: | a x + b | = c, where c > 0, is equivalent to the compound statement a x + b = c or a x + b = - c

The value inside the absolute symbols has 2 solutions, + and – ,

| a x + b = c or a x + b = - c | = c

But, remember that Absolute Value is always positive and has 2 solutions. “Distance has to be +”, so in this example the value of c outside the absolute value symbols is + even when one of the solutions inside the absolute value is negative. See also example above, where c = + 3.

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Solving Absolute Value Equations 1.6

Transformations of absolute value inequalities

| a x + b | < c, where c > 0 means that a x + b is between – c and c, - c < a x + b < c

[compound statement “and”, where both conditions must be met.]

| a x + b | > c, where c > 0 means that a x + b is beyond – c and c, - c > a x + b or a x + b > c

[two statements “or”, where only one condition must be met.]

Remember: Absolute Value means distance ( + value) and always has 2 solutions.

| a x + b | < c means “and”, - c < a x + b < c, where a x + b is between – c and + c

| a x + b | > c means “or”, - c > a x + b or a x + b > c, where a x + b is beyond – c and + c

- c + c

- c + c

<

>32

Solving Absolute Value Inequalities 1.6

Example: To solve an absolute value equation, rewrite it as two linear equations. To solve an absolute value inequality, rewrite it as a compound inequality.

| x + 3 | = 5

x + 3 = 5 x + 3 = – 5

x = 2 x = – 8

| x – 7 | 2

– 2 x – 7 2

+ 7 + 7 + 7

5 x 9

5 9

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Solving Absolute Value Expressions and Inequalities

1.7