# 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

- 6th Grade IAR Math Worksheets: FREE & Printable
- 3rd Grade SC Ready Math Worksheets: FREE & Printable
- 7th Grade PSSA Math FREE Sample Practice Questions
- Using Algebra Tiles to Model and Solve Equations
- Discontinuous Function
- How to Write a Point-slope Form Equation from a Graph?
- Full-Length 6th Grade STAAR Math Practice Test
- Hyperbola in Standard Form and Vertices, Co– Vertices, Foci, and Asymptotes of a Hyperbola
- What Does the CBEST Test Qualify You For?
- Picture Perfect Inequalities: How to Graph Solutions of Two-Step Inequalities

## What people say about "How to Solve an Absolute Value Inequality? - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.