How to Solve an Absolute Value Inequality?

The absolute value of \(a\) is written as \(|a|\). For any real numbers \(a\) and \(b\), if \(|a| < b\), then \(a < b\) and \(a > -b\) and if \(|a| > b\), then \(a > b\) and \(a < -b\).

Related Topics

• How to Solve Absolute Value Equations
• How to Graph Absolute Value Inequalities

A step-by-step guide to solving an absolute value inequality

To solve an absolute value inequality, follow the below steps:

• Isolate the absolute value expression.
• Write the equivalent compound inequality.
• Solve the compound inequality.

Solving Absolute Value Inequalities – Example 1:

Solve \(|x-5|<3\).

Solution:

To solve this inequality, break it into a compound inequality: \(x-5<3\) and \(x-5>-3\)

So, \(-3<x-5<3\).

Add \(5\) to each expression: \(-3+5<x-5+5<3+5 → 2<x<8\).

Solving Absolute Value Inequalities – Example 2:

Solve \(|x+4| ≥ 9\).

Solution:

Split into two inequalities: \(x+4 ≥ 9\) or \(x+4 ≤ -9\).

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Subtract \(4\) from each side of each inequality:

\(x+4-4 ≥ 9-4\) → \(x ≥ 5\)

or

\(x+4-4 ≤ -9-4\) → \(x ≤ -13\)

Exercises for Absolute Value Inequalities

Solve each absolute value inequality.

• \(\color{blue}{|4x|<12}\)

• \(\color{blue}{|x-5|>9}\)

• \(\color{blue}{|3x-7|<8}\)

• \(\color{blue}{5|x-2|>20}\)

• \(\color{blue}{-3<x<3}\)
• \(\color{blue}{x< -4 \:or\: x>14}\)
• \(\color{blue}{-\frac{1}{3}<x<5}\)
• \(\color{blue}{x<-2 \:or\: x>6}\)