CHAPTER 6Section 6.4 Solving Absolute Value Equations and Inequalities

SOLVE ABSOLUTE VALUE INEQUALITIES

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93 x842 x

4x All the numbers whose distance from zero is greater than 4.

-4 4

4x 4xor

***Notice that you need to have two inequalities to represent the distances that are greater than 4 from zero.

What does really mean? 4x

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“greatOR”• If the inequality involves a > or ≥, it is an “OR” inequality.

4x All the numbers whose distance from zero is less than 4.

4x 4x-4 4

and

***Notice that you need to have two inequalities to represent the distances that are less than 4 from zero.

However, since these inequalities must happen at the same time, it should be written as

44 x

What does really mean? 4x

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“less thAND”• If the inequality involves a < or ≤, it is an “AND” inequality.

Solve and graph

Step 1: Isolate the absolute value.

842 x

44 x

44 x 44 x

8x 0x

0 8

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Step 2: Set up two inequalities.

Step 3: Solve the

inequalities.

Step 4: Graph the solutions.

Let’s practice!

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You Try!Step 1: Solve and graph.

Step 2: Step 3:

Step 4:

Step 5:

11124 x

32 x

32 x 32 x

1x 5x

-5 1

Final answer: -5<x<1© 2011 The Enlightened Elephant

You Try!

Solve and graph.

93 x

93 x 93 x

12x6x

-12 6

Final answer: 12x6x or

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Set up two inequalities!

You Try! Solve and graph.

21123 x

712 x 712 x

3x4x

-3 4

Final answer: 3x4x or

712 x

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Isolate the absolute value

first!

You Try! Solve and graph.

15363 x

1263 x 1263 x

2x6x

Final answer: 62 x

1263 x

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Isolate the absolute value

first!

-2 6

You Try! Solve and graph.

238123 x

512 x 512 x

3x2x

Final answer: 23 x

Isolate the absolute value

first!

512 x

-3 2

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You Try! Solve and graph.

1282 x

Final answer: NO SOLUTION!

Absolute values are distances, which cannot be negative.

You can choose any value for x and the left side of the inequality will

always be positive. However, positive numbers are

NEVER less than negative numbers.© 2011 The Enlightened Elephant

Isolate the absolute value

first!

You Try! Solve and graph.

10126 x

Final answer: ALL REAL NUMBERS!

Isolate the absolute value

first!

Absolute values are distances, which cannot be negative.

You can choose any value for x and the left side of the inequality will

always be positive. Positive numbers are ALWAYS greater

than negative numbers.

© 2011 The Enlightened Elephant

Success!Nice job!

© 2011 The Enlightened Elephant