Chapter 2.7 – Absolute Value Inequalities

Objectives

Solve absolute value inequalities of the form /x/ < a

Solve absolute value inequalities of the form /x/ > a

Example 1

Solve: ￨ x ￨ 3

[ ]

The solution set is {-3, 3}

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

Solving Absolute Value expression of the form ￨ x ￨ < a If a is a positive number, then ￨ x ￨ < a

is equivalent to –a < x< a

Example 2:

Solve for m: ￨m – 6 ￨ < 2

Step 1: -a < x < a

Replace x with m - 6 and a with 2

-2 < m – 6 < 2

Example 2 continued

Solve the compound inequality

-2 < m – 6 < 2

-2 + 6 < m – 6 + 6 < 2 + 6

4 < m < 8

The solution set is (4, 8) and its graph is,

Give it a try!

Solve ￨ x-2 ￨ 1

HINT

Before using an absolute value inequality property, you MUST MUST ISOLATE the absolute value ISOLATE the absolute value expressionexpression on one side of the inequality!

Example 3:

Solve for x: ￨ 5x + 1 ￨ + 1 10

Give it a try!

Solve: ￨ 2x - 5 ￨ + 2 9

Example 4

Solve for x:

The absolute value of a number is always nonnegative and can never be less than –13. This inequality has NO solution! The solution set is { } or 0

1310

12 x

Give it a try!

Solve: 47

25 x

Let ￨ x ￨ 3

The solution set includes all numbers who distance from 0 is 3 or more units. The graph of the solution set contains 3 and all points to the right of 3 on the number line or –3 and all points to the left of –3 on the number line..

The solution is ( - , -3] U [3, )

Form of ￨ x ￨ > a

Solving Absolute Value Inequalities of the form ￨ x ￨ > a If a is a positive number, the ￨ x ￨ > a

is equivalent to x < -a OR x > a

Example 5

Solve for y:

Step 1: Rewrite the inequalities without absolute value bars

y – 3 < -7 or y – 3 > 7Step 2: Solve the compound inequality

y < -4 or y > 10

3 7y

Example 5 Continued

Step 3: Graph the inequalities

Step 4: Write the solution set

(-∞, -4) U (10, ∞)

Give it a try!

Solve: 2 4x

Example 6: Isolate the Absolute Value expression!

Solve:

Step 1: Isolate the Absolute Value Expression

2 9 5 3x

52 59 5 3x

2 9 2x

Example 6 - Continued

***Remember***

The absolute value of any number is always a nonnegative and thus is always great than -2. The inequality and the original inequality are true for all values of x. The solution set is {x/ x is a real number} or (-∞,∞)

Example 7: Isolate the Absolute Value Expression!

Solve:

Step 1: Isolate the Absolute Value Expression

1 7 53x

73

71 7 5x

13

2x

Example 7 - Continued

Step 2: Write the absolute value inequality as an equivalent compound inequality.

1 2 1 23 3

1 33 3

3 9

x xor

x xor

x or x

Example 7 continued

Step 3: Solve each inequality

1 2 1 23 3

1 33 3

3 9

x xor

x xor

x or x

Example 7 continued

Step 4: Write the solution set and graph

(-∞, -3] U [9, ∞)

Give it a try!

Solve: 3 8 45x

Example 8 - Zero

Solve for x:

**Remember – the absolute value of any expression will never be less than 0, but it may be equal to 0. Thus solve the equation by setting it equalequal to zero!

2 10

3

x

Example 8 Continued 2 1

03

x

2( 1)0

3

x

2( 1)0

33 3

x

2 2 0x 2 2x

1x

You give try!

Solve: 5( 3)

02

x

Chart on Page 113

Copy the chart from 113 into your notes!