How to Solve an Absolute Value Inequality?

The absolute value of inequalities follows the same rules as the absolute value of numbers.

How to Solve an Absolute Value Inequality?

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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Step by step guide to solve an absolute value inequality

To solve an absolute value inequality, follow 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 sides 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 inequalities.

  • \(\color{blue}{|4x|<12}\)
  • \(\color{blue}{|x-5|>9}\)
  • \(\color{blue}{|3x-7|<8}\)
  • \(\color{blue}{5|x-2|>20}\)
This image has an empty alt attribute; its file name is answers.png
  • \(\color{blue}{-3<x<3}\)
  • \(\color{blue}{x< -4 \:or\: x>14}\)
  • \(\color{blue}{-\frac{1}{3}<x<5}\)
  • \(\color{blue}{x<-2 \:or\: x>6}\)

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