# 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 \(a < -b\).

## Related Topics

## 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.**

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

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

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

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

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

## Related to This Article

### More math articles

- Can You Cheat on the ALEKS Test?
- 4th Grade North Carolina End-of-Grade Math Worksheets: FREE & Printable
- Full-Length 7th Grade PSSA Math Practice Test
- Place Value Word Problems
- How to Prepare for the TSI Math Test?
- 5 Essential Strategies in Teaching Math
- How to Pass TSI Test: Top Tips and Key Tactics
- Best Laptops for Online Math Teaching
- Half-Angle Identities
- How to Graph the Cosecant Function?

## What people say about "How to Solve an Absolute Value Inequality?"?

No one replied yet.