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# 6.6: Absolute Value Inequalities

Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

• Solve absolute value inequalities.
• Rewrite and solve absolute value inequalities as compound inequalities.
• Solve real-world problems using absolute value inequalities.

## Vocabulary

Terms introduced in this lesson:

absolute value inequality

## Teaching Strategies and Tips

Use Example 1 to show that the distance interpretation equally applies to absolute value inequalities.

a. Solve the inequality.

$|x| \le 10.$

Solution: $|x| \le 10$ represents all numbers whose distance from the origin is less than or equal to $10$. This means that $-10 \le x \le 10$.

b. Solve the inequality.

$|x| \ge 10.$

Solution: $|x| \ge 10$ represents all numbers whose distance from the origin is greater than or equal to $10$. This means that $x \le -10$ or $x \ge 10$.

Use Example 1 and the distance interpretation to motivate solving the absolute value inequalities in Examples 2-5.

• Allow students to infer from Example 1 that $|x| < a \Leftrightarrow -a < x < a$ and $|x| > a \Leftrightarrow x < -a$ or $x > a$.

In Problems 4 and 5 in the Review Questions, have students divide by the coefficient of $x$ first.

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Feb 22, 2012

Aug 22, 2014