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
- The Ultimate AZMerit Algebra 1 Course (+FREE Worksheets)
- Series Behavior with the n-th Term Test: Divergence Test
- 4th Grade NDSA Math Worksheets: FREE & Printable
- How is the FTCE General Knowledge Test Scored?
- What Does ALEKS Stand for?
- How to Solve and Graph One-Step Multiplication and Division Equations
- Full-Length ACT Math Practice Test
- Top 10 Free Websites for SHSAT Math Preparation
- Top 10 7th Grade STAAR Math Practice Questions
- 3rd Grade FSA Math Worksheets: FREE & Printable
What people say about "How to Solve an Absolute Value Inequality? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.