Module 1 ~ Topic 4Solving Absolute Value Inequalities

Table of Contents

Slides 2-3: How to Solve Absolute Value Inequalities Slides 4-5: How to write the answer appropriately Slide 6: Rules Slides 7-10: Examples Slide 11: Practice Problems

Audio/Video and Interactive Sites

Slide 12: Interactive

To solve Absolute Value Inequalities1. Isolate the absolute value expression on one side.

a) To do this, do any addition or subtraction firstb) Then do any multiplication or division last. ** If you multiply or

divide by a negative number, make sure you flip the inequality sign in this step. 2.

a) Set up two inequalities if the absolute value expression is opposite a 0 or a positive number:

For the first equation, rewrite the expression that is inside the bars (but do not use the bars) and set it up using the same inequality sign that is in the original problem.

For the second equation, rewrite the expression that is inside the bars again (but do not use the bars) and set it up using the opposite inequality sign of the one in the original problem.

WRITE THE ANSWER IN CORRECT FORM

b) If the isolated absolute value expression is set up to be less than a negative number, there is no solution. You can stop.

Absolute Value Inequalities Absolute Value Inequalities

Explanation and Examples

There are several rules that apply to absolute value inequalities.

You must be very careful when solving and you also must restate your answers correctly before submitting your assignments.

Two Types of AnswersTwo Types of Answers

AND OR

“AND” problems include all numbers between the two solutions found on the number line.

“OR” problems include all numbers in opposite directions on the number line.

0A AB B

BxA Bxorx A

Two Types of AnswersTwo Types of Answers

AND OR

0A AB B

BxA Bxorx A

Notice, visually, how the answer resembles the graph.

All answers (x) are Between A AND B. Notice: The word “and” is NOT in the answer.

All answers, x, are either less than A OR greater than B. Notice the word “or’ is in the answer.

C x toequivalent x then positive, is C If C

C or x C - x as same theis x 4)

C or x C - x as same theis x 3)

C x C - as same theis x 2)

C x C - as same theis x 1)

: thenpositive, is C If

C

C

C

C

C

C

C

C

x inequality thesatisfiesnumber realEvery )4

x inequality thesatisfiesnumber realEvery 3)

solution no has x inequality The 2)

solution no has x inequality The 1)

thennegative, is C If

0x inequality thesatisfiesnumber realEvery 4)

0 or x 0 x as same theis 0x inequality The 3)

0 x is 0x inequality theofsolution The 2)

solution no has 0x inequality The )1

CasesSpecial

Examples: Absolute Value Examples: Absolute Value Inequalities Inequalities

8x

8x 8x

0-8 8

The solution to | x | > 8 is: x < -8 or x > 8The solution to | x | > 8 is: x < -8 or x > 8

Use a number line to help you with the solution.

Use a number line to help you with the solution.

84 x

84 x 84 x Notice: You will be dividing by a negative number, don’t forget to flip the sign!!

2x 2x

0-2 2

The solution to | -4x | > 8 is: x < -2 or x > 2The solution to | -4x | > 8 is: x < -2 or x > 2

Use a number line to help you with the solution.

Use a number line to help you with the solution.

2794 x

Notice: You will be dividing by a negative number, don’t forget to flip the sign!!

Notice: You will be dividing by a negative number, don’t forget to flip the sign!!

2794 x

2794 x 2794 x

364 x 184 x

9x2

9x

-9 02

9

2

99 xThe solution isThe solution is

Use a number line to help you with the solution.

Use a number line to help you with the solution.

07

12 x

Since the absolute value is shown to be less than 0, this problem has a solution of “No Solution”.

Solve the Problem Solution Solution in Interval Form

129 x9

1

3

1 x

9

1,

3

1

523 m 31 x 3,1

104 p 2812 x 28,12

1062 k3

4 2 kork

,2

3

4, or

4812 m 1 3

1 morm 1,

3

1,

or

More Explanation and Examples

More Explanation and Examples

Interactive Practice Problems