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Module 1 ~ Topic 4
Solving 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 Inequalities
1. Isolate the absolute value expression on one side.
a) To do this, do any addition or subtraction first
b) 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
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 Answers
AND
“AND” problems include all
numbers between the two
solutions found on the
number line.
A
B
A x B
OR
“OR” problems include all
numbers in opposite
directions on the number line.
A
0
B
x  A or x  B
Two Types of Answers
AND
OR
Notice, visually, how the answer resembles the graph.
A
B
A x B
All answers (x) are
Between A AND B.
Notice: The word “and” is
NOT in the answer.
A
0
B
x  A or x  B
All answers, x, are either
less than A OR greater
than B. Notice the word
“or’ is in the answer.
If C is positive, then x  C equivalent to x  C
If C is positive, then :
1) x  C is the same as - C  x  C
2) x  C is the same as - C  x  C
3) x  C is the same as x  - C or x  C
4) x  C is the same as x  - C or x  C
If C is negative, then
1) The inequality x  C has no solution
2) The inequality x  C has no solution
3) Every real number satisfies the inequality x  C
4) Every real number satisfies the inequality x  C
Special Cases
1) The inequality x  0 has no solution
2) The solution of the inequality x  0 is x  0
3) The inequality x  0 is the same as x  0 or x  0
4) Every real number satisfies the inequality x  0
Examples: Absolute Value Inequalities
x 8
x  8
x 8
-8
The solution to | x | > 8 is:
0
8
x < -8 or x > 8
Use a number line to help
you with the solution.
4 x  8
4 x  8
4 x  8
x  2
-2
x2
0
The solution to | -4x | > 8 is:
Notice: You will be dividing by a
negative number, don’t forget to
flip the sign!!
Use a number line to help
you with the solution.
2
x < -2 or x > 2
Notice: You will be dividing by a
negative number, don’t forget to
flip the sign!!
 4 x  9  27
4 x  9  27
4 x  9  27
4 x  36
4 x  9  27
4 x  18
9
x
2
x  9
-9
0
Notice: You will be dividing by a
negative number, don’t forget to
flip the sign!!
Use a number line to help
you with the solution.
9
2
The solution is
9
9 x 
2
12
x 0
7
Since the absolute value is shown to be
less than 0, this problem has a solution of
“No Solution”.
Solve the Problem
Solution
9x  2  1

12m  8  4
p  4  10
2  6k  10
3  2m  5
m
Solution in Interval Form
1
1
x
3
9
 1 1
  3 , 9 


1

  ,  or 1,  
3

1
or m  1
3
12,28
12  x  28
k  2 or k  
1  x  3
4
3
4

  ,  or 2,  
3

1,3
If you are unable to get these answers, please ask your instructor for help
before attempting the assignments.
More Explanation and Examples
More Explanation and Examples
Interactive Practice Problems