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
- 7th Grade RICAS Math Worksheets: FREE & Printable
- Top 10 ALEKS Math Prep Books (Our 2023 Favorite Picks)
- How to Multiply Three or More Numbers?
- FREE 3rd Grade Common Core Math Practice Test
- How to Graph Quadratic Functions?
- How to Work with the Intermediate Value Theorem?
- Top 10 6th Grade MEAP Math Practice Questions
- Can you Teach in California without a Credential?
- Fundamental Theorem of Calculus: A Principle That Saves Your Life
- 10 Most Common 4th Grade OST Math Questions
What people say about "How to Solve an Absolute Value Inequality? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.