Chapter 5Linear Inequalities

Section 5-1Solving Inequalities by Addition and Subtraction

Essential Question

How do you solve and graph inequalities with addition and subtraction?

Vocabulary

1. Set-builder notation:

Vocabulary

1. Set-builder notation: A mathematical way of writing the solution to an inequality

Vocabulary

1. Set-builder notation: A mathematical way of writing the solution to an inequality

x | x ≤ 7{ }

Vocabulary

1. Set-builder notation: A mathematical way of writing the solution to an inequality

x | x ≤ 7{ }

“The set of x such that x is less than or equal to seven”

Notations

<

>

≤

≥

Notations

<

>

≤

≥

“Less than”

Notations

<

>

≤

≥

“Less than” Open end point

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than”

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than” Open end point

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than” Open end point Shade to the right

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than” Open end point Shade to the right

“Less than or equal to”

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than” Open end point Shade to the right

“Less than or equal to” Closed end point

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than” Open end point Shade to the right

“Less than or equal to” Closed end point Shade to the left

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than” Open end point Shade to the right

“Less than or equal to” Closed end point Shade to the left

“Greater than or equal to”

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than” Open end point Shade to the right

“Less than or equal to” Closed end point Shade to the left

“Greater than or equal to” Closed end point

Notations

<

>

≤

≥

“Less than” Open end point Shade to the left

“Greater than” Open end point Shade to the right

“Less than or equal to” Closed end point Shade to the left

“Greater than or equal to” Closed end point Shade to the right

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12 +12

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12 +12

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12 +12

c > 77

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12 +12

c > 77

c | c > 77{ }

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12 +12

c > 77

c | c > 77{ }

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12 +12

c > 77

c | c > 77{ }

77 78 79 80 8176757473

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12 +12

c > 77

c | c > 77{ }

77 78 79 80 8176757473

Example 1Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

c −12 > 65 +12 +12

c > 77

c | c > 77{ }

77 78 79 80 8176757473

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23 −23

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23 −23

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23 −23

x < −9

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23 −23

x < −9

x | x < −9{ }

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23 −23

x < −9

x | x < −9{ }

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23 −23

x < −9

x | x < −9{ }

-9 -8 -7 -6 -5-10-11-12-13

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23 −23

x < −9

x | x < −9{ }

-9 -8 -7 -6 -5-10-11-12-13

Example 2Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

x + 23 <14 −23 −23

x < −9

x | x < −9{ }

-9 -8 -7 -6 -5-10-11-12-13

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −13n

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −13n −13n

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −13n −13n

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −n ≤ 4

−13n −13n

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −n ≤ 4

−13n −13n

−1

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −n ≤ 4

−13n −13n

−1 −1

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −n ≤ 4

n | n ≥ −4{ }

−13n −13n

−1 −1

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −n ≤ 4

n | n ≥ −4{ }

−13n −13n

−1 −1

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −n ≤ 4

n | n ≥ −4{ }

-4 -3 -2 -1 0-5-6-7-8

−13n −13n

−1 −1

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −n ≤ 4

n | n ≥ −4{ }

-4 -3 -2 -1 0-5-6-7-8

−13n −13n

−1 −1

Example 3Solve the inequality. Write your solution in set-builder notation.

Graph your solution.

12n − 4 ≤13n +4 +4 −n ≤ 4

n | n ≥ −4{ }

-4 -3 -2 -1 0-5-6-7-8

−13n −13n

−1 −1

Example 4

Matt Mitarnowski wants to buy season passes to two theme parks. If one season pass is $54.99 (tax included) and he has $100 to spend on both

passes, the second season pass must cost no more than what amount?

Example 4

Matt Mitarnowski wants to buy season passes to two theme parks. If one season pass is $54.99 (tax included) and he has $100 to spend on both

passes, the second season pass must cost no more than what amount?

Let x be the cost of the second ticket

Example 4

Matt Mitarnowski wants to buy season passes to two theme parks. If one season pass is $54.99 (tax included) and he has $100 to spend on both

passes, the second season pass must cost no more than what amount?

Let x be the cost of the second ticket

x +54.99 ≤100

Example 4

Matt Mitarnowski wants to buy season passes to two theme parks. If one season pass is $54.99 (tax included) and he has $100 to spend on both

passes, the second season pass must cost no more than what amount?

Let x be the cost of the second ticket

x +54.99 ≤100 −54.99

Example 4

Matt Mitarnowski wants to buy season passes to two theme parks. If one season pass is $54.99 (tax included) and he has $100 to spend on both

passes, the second season pass must cost no more than what amount?

Let x be the cost of the second ticket

x +54.99 ≤100 −54.99 −54.99

Example 4

Matt Mitarnowski wants to buy season passes to two theme parks. If one season pass is $54.99 (tax included) and he has $100 to spend on both

passes, the second season pass must cost no more than what amount?

Let x be the cost of the second ticket

x +54.99 ≤100 −54.99 −54.99

Example 4

Matt Mitarnowski wants to buy season passes to two theme parks. If one season pass is $54.99 (tax included) and he has $100 to spend on both

passes, the second season pass must cost no more than what amount?

Let x be the cost of the second ticket

x +54.99 ≤100 −54.99 −54.99

x ≤ 45.01

Example 4

Matt Mitarnowski wants to buy season passes to two theme parks. If one season pass is $54.99 (tax included) and he has $100 to spend on both

passes, the second season pass must cost no more than what amount?

Let x be the cost of the second ticket

x +54.99 ≤100 −54.99 −54.99

x ≤ 45.01

The second ticket must be $45.01 or less

Summarizer

Compare and contrast the following graphs.

a < 4 a ≤ 4

Problem Sets

Problem Sets

Problem Set 1: p. 286 #1-11

Problem Set 2: p.286 #12-27 multiples of 3, #30-40 all

"I have found power in the mysteries of thought." - Euripides