Copyright © 2013 Pearson Education, Inc.

Section 2.5

Linear Inequalities

Solutions and Number Line Graphs

A linear inequality results whenever the equals sign in a linear equation is replaced with any one of the symbols <, ≤, >, or ≥.

x > 5, 3x + 4 < 0, 1 – y ≥ 9

A solution to an inequality is a value of the variable that makes the statement true. The set of all solutions is called the solution set.

Page 136

Example

Use a number line to graph the solution set to each inequality.

a.

b.

c.

d.

1x

1x

3 x

2x

Page 137

31 .

1 b.

3 .

xc

x

xa

Linear Inequalities in One Variable

)

[

](

Page 137

Interval Notation

Each number line graphed on the previous slide represents an interval of real numbers that corresponds to the solution set to an inequality.Brackets and parentheses can be used to represent the interval. For example:

1x (1, )

1x [1, )

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Example Interval Notation

Write the solution set to each inequality in interval notation.

a. b.

Solution

a. b.

More examples

6x 2y

(6, ) ( , 2]

5

0.

x

x

d.

c

)5,(

),0[

Page 137

Example Checking a Solution

Determine whether the given value of x is a solution to the inequality.Solution

4 2 8, 7x x

?

?

4 2 <8

4( 87) 2

x

?

28 2 8 ?

26 8 e Fals

Page 138

The Addition Property of InequalitiesPage 139

Example

Solve each inequality. Then graph the solution set.a. x – 2 > 3 b. 4 + 2x ≤ 6 + xSolution

a. x – 2 > 3

x – 2 + 2 > 3 + 2

x > 5

b. 4 + 2x ≤ 6 + x

4 + 2x – x ≤ 6 + x – x

4 + x ≤ 6

4 – 4 + x ≤ 6 – 4

x ≤ 2

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96 x

Properties of Inequalities

)

[

)3,(

),2[

6 6 3x }3|{ xx

4728 xx2 2

278 xx7x-7x - 2x

}2|{ xx

Page 140

The Multiplication Property of InequalitiesPage 141

Example

Solve each inequality. Then graph the solution set.a. 4x > 12 b.Solutiona. 4x > 12

12

4x

4 2

4 4

1x

3x

b. 1

24x

4 (1

( 2)4

4) x

8 x

Page 141

Example

Solve each inequality. Write the solution set in set-builder notation.a. 4x – 8 > 12 b. Solutiona. 4x – 8 > 12

4 3 4 5x x

4 8 8 12 8x 4 20x

5x

{ | 5}x x

b. 4 3 4 5x x

4 3 3 4 5 3x x x x

4 5x

4 5 5 5x

9 x

{ | 9}x x

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24

1x

Properties of Inequalities

)8,(

8x }8|{ xx

4 4)

Page 142

186 x(

),3( 6 6

3x

Sign changes

2536 xx

Linear Inequalities

[

),1[

8 8

1

1

x

x

Add 2

xx 538 Add 3x

x88

same

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2536 xx853 xx

Subtract 6

Subtract 5x

88 x8- 8-

1xSign changes

14)2(31)3(2 xx

Linear Inequalities

1463162 xx

8372 xx Add 8

xx 312 Sub 2x

1

1

x

x

[

),1[

Page 142

Applications

To solve applications involving inequalities, we often have to translate words to mathematical statements.

Page 143

Example

Translate each phrase to an inequality. Let the variable be x.a. A number that is more than 25.

b. A height that is at least 42 inches.

x > 25

x ≥ 42

Page 143

Example

For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250.

a. Write a formula that gives the cost C of producing x cases of snacks.

b. Write a formula that gives the revenue R from selling x cases of snacks.

C = 135x + 175,000

R = 250x

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Example (cont)

For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250.

c. Profit equals revenue minus cost. Write a formula that calculates the profit P from selling x cases of snacks.

P = R – C

= 250x – (135x + 175,000)

= 115x – 175,000

Page 144

Example (cont)

For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250.

d. How many cases need to be sold to yield a positive profit?

115x – 175,000 > 0

115x > 175,000

x > 1521.74

Must sell at least 1522 cases.

Page 144

DONE

Objectives

• Solutions and Number Line Graphs

• The Addition Property of Inequalities

• The Multiplication Property of Inequalities

• Applications

EXAMPLEGraphing inequalities on a number line

Use a number line to graph the solution set to each inequality.

a.

b.

c.

d.

1x

1x

5x

2x