How to Solve an Absolute Value Inequality?
The absolute value of inequalities follows the same rules as the absolute value of numbers.
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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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