Solve Absolute Value Inequalities
Goal • Solve absolute value inequalities.
Example 1 Solve an absolute value inequality
Solve the inequality. Graph your solution.
a. x ≤ 9
b. x > 1/4
Solution
a. The distance between x and 0 is less than or equal
to 9. So, 9 ≤ x ≤ 9 . The solutions are all real
numbers less than or equal to 9 and greater than
or equal to 9 .
b. The distance between x and 0 is greater than 1/4.
So, x > 1/4 or x < 1/4. The solutions are all real
numbers greater than
1/4
or less than 1/4.
SOLVING ABSOLUTE VALUE INEQUALITIES
• The inequality ax + b< c where c > 0 is equivalent
to the compound inequality c < ax + b < c.
• The inequality ax + b> c where c > 0 is equivalent
to the compound inequality ax + b < c or ax + b > c.
These statements are also true for inequalities involving
≥ and ≤.
Example 2 Solve an absolute value inequality
Solve 2x  7< 9. Graph your solution.
2x  7< 9 
Write original inequality. 
9 < 2x  7 < 9 
Rewrite as compound inequality. 
2 < 2x < 16 
Add 7 to each expression. 
1 < x < 8 
Divide each expression by 2 . 
The solutions are all real numbers greater than 1
and less than 8 . Check several solutions in the original
inequality.
Example 3 Multiple Choice Practice
What is the solution of the inequality x + 8  4 ≥ 2?
Solution
x + 8  4 ≥ 2 
Write original
inequality. 
x + 8≥ 6 
Add 4 to
each side. 
x + 8 ≥ 6 or x + 8 ≤ 6 
Rewrite as
compound
inequality. 
x ≥ 2 or x ≤ 14 
Subtract 8
from each side. 
The correct answer is D . A B C D
• Guided Practice Solve the inequality. Graph your
solution.
SOLVING INEQUALITIES
OneStep and MultiStep Inequalities
• Follow the steps for solving an equation, but reverse
the inequality symbol when multiplying or dividing
by a negative number .
Compound Inequalities
• If necessary, rewrite the inequality as two separate
inequalities. Then solve each inequality separately.
Include and or or in the solution.
Absolute Value Inequalities
• If necessary, isolate the absolute value expression on
one side of the inequality. Rewrite the absolute value
inequality as a compound inequality . Then solve
the compound inequality.
