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6.6: Absolute Value Inequalities

Difficulty Level: At Grade Created by: CK-12
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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*}|x|10.

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

b. Solve the inequality.

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

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

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

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

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CK.MAT.ENG.TE.1.Algebra-I.6.6