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

One-Step and Multi-Step 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.