# How to Solve an Absolute Value Inequality?

The absolute value of inequalities follows the same rules as the absolute value of numbers.

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 \(a < -b\).

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## 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 \(x+4 ≤ -9\).

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.**

- \(\color{blue}{|4x|<12}\)

- \(\color{blue}{|x-5|>9}\)

- \(\color{blue}{|3x-7|<8}\)

- \(\color{blue}{5|x-2|>20}\)

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

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