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# 6.6: Absolute Value Inequalities

Difficulty Level: 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.

a. Solve the inequality.

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

Solution: |x|10\begin{align*}|x| \le 10\end{align*} represents all numbers whose distance from the origin is less than or equal to 10\begin{align*}10\end{align*}. This means that 10x10\begin{align*}-10 \le x \le 10\end{align*}.

b. Solve the inequality.

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

Solution: |x|10\begin{align*}|x| \ge 10\end{align*} represents all numbers whose distance from the origin is greater than or equal to 10\begin{align*}10\end{align*}. This means that x10\begin{align*}x \le -10\end{align*} or x10\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 |x|<aa<x<a\begin{align*}|x| < a \Leftrightarrow -a < x < a\end{align*} and |x|>ax<a\begin{align*}|x| > a \Leftrightarrow x < -a\end{align*} or x>a\begin{align*}x > a\end{align*}.

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

## Error Troubleshooting

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