# 6.6: Absolute Value Inequalities

**At Grade**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*.

Additional Examples:

a. *Solve the inequality.*

\begin{align*}|x| \le 10.\end{align*}

Solution: \begin{align*}|x| \le 10\end{align*} represents all numbers whose distance from the origin is *less than or equal to* \begin{align*}10\end{align*}. This means that \begin{align*}-10 \le x \le 10\end{align*}.

b. *Solve the inequality.*

\begin{align*}|x| \ge 10.\end{align*}

Solution: \begin{align*}|x| \ge 10\end{align*} represents all numbers whose distance from the origin is *greater than or equal* to \begin{align*}10\end{align*}. This means that \begin{align*}x \le -10\end{align*} or \begin{align*}x \ge 10\end{align*}.

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 \begin{align*}|x| < a \Leftrightarrow -a < x < a\end{align*} and \begin{align*}|x| > a \Leftrightarrow x < -a\end{align*} or \begin{align*}x > a\end{align*}.

In Problems 4 and 5 in the *Review Questions*, have students divide by the coefficient of \begin{align*}x\end{align*} first.

## Error Troubleshooting

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