How to Solve an Absolute Value Inequality?

Solving absolute value inequalities follows two distinct rules depending on whether the inequality is a “less than” or “greater than” type. Once you know which rule to apply, the process reduces to solving a familiar compound inequality and graphing the solution on a number line. This guide explains both rules step by step, with worked examples, two video lessons, and practice problems.

What Is an Absolute Value Inequality?

An absolute value inequality is an inequality that contains an absolute value expression, such as \(\color{blue}{|x – 3| < 5}\) or \(\color{blue}{|2x + 1| \ge 9}\). Because absolute value measures distance from zero, solving these inequalities means finding all values that are either within a certain distance from a number (“less than” type) or beyond that distance (“greater than” type).

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Two Rules for Solving Absolute Value Inequalities

Rule 1: “Less Than” Type (< or ≤) → Compound AND

If \(\color{blue}{|\text{ expression }| < c}\) (where c > 0), rewrite as:

−c < expression < c

This is a single compound inequality. Solve it to find a bounded interval.

Quick example: |x| < 4 → −4 < x < 4

Rule 2: “Greater Than” Type (> or ≥) → Compound OR

If \(\color{blue}{|\text{ expression }| > c}\) (where c > 0), rewrite as:

expression < −c   OR   expression > c

This gives two separate inequalities. Solve each one. The solution is the union of both.

Quick example: |x| > 4 → x < −4 or x > 4

Special Cases

• If |expression| < c and \(\color{blue}{c \le 0}\): No solution (absolute value is \(\color{blue}{\text{ always } \ge 0}\), so it cannot be less than a non-positive number).
• If |expression| > c and c < 0: All real numbers (absolute value is \(\color{blue}{\text{ always } \ge 0}\), which is always > any negative number).

Step-by-Step Summary

1. Isolate the absolute value expression on one side of the inequality.
2. Check the right side. If negative with “<”: no solution. If negative with “>”: all reals.
3. “Less than” (< or ≤): \(\color{blue}{\text{ write } -c}\) < expression < c and solve the compound inequality.
4. “Greater than” (> or ≥): write two inequalities (expression < −c OR expression > c) and solve each.
5. Write the solution in inequality notation or interval notation and graph on a number line.

Watch: Solving Absolute Value Inequalities (Video Lesson)

The Organic Chemistry Tutor explains the “and” and “or” rules with clear examples:

Solving Absolute Value Inequalities – Worked Examples

Example 1: Solve \(\color{blue}{|x – 3| < 5}\)

“Less than” → compound AND:
−5 < \(\color{blue}{x – 3}\) < 5
Add 3 throughout: −\(\color{blue}{5 + 3}\) < x < \(\color{blue}{5 + 3}\)
−2 < x < 8

Example 2: Solve \(\color{blue}{|2x + 1| \ge 9}\)

“Greater than or equal” → OR:
\(\color{blue}{2x + 1}\) ≤ −9   OR   \(\color{blue}{2x + 1 \ge 9}\)
Case 1: 2x ≤ −10 → x ≤ −5
Case 2: \(\color{blue}{2x \ge 8}\) → \(\color{blue}{x \ge 4}\)
x ≤ −5 or \(\color{blue}{x \ge 4}\)

Example 3: Solve \(\color{blue}{|x + 4| \le 7}\)

“Less than or equal” → compound AND:
−\(\color{blue}{7 \le x + 4 \le 7}\)
Subtract 4: −\(\color{blue}{7 – 4 \le x \le 7 – 4}\)
−\(\color{blue}{11 \le x \le 3}\)

Example 4: Solve \(\color{blue}{|3x – 6| > 12}\)

“Greater than” → OR:
\(\color{blue}{3x – 6}\) < −12   OR   \(\color{blue}{3x – 6}\) > 12
Case 1: 3x < −6 → x < −2
Case 2: 3x > 18 → x > 6
x < −2 or x > 6

More Practice: Absolute Value Inequalities – Basic Introduction (Video Lesson)

The Organic Chemistry Tutor provides additional worked examples including graphing on number lines:

Exercises: Solving Absolute Value Inequalities

1. \(\color{blue}{|x| < 6}\)
2. \(\color{blue}{|x + 2| > 5}\)
3. \(\color{blue}{|3x – 9| \le 12}\)
4. \(\color{blue}{|2x + 4| \ge 10}\)
5. \(\color{blue}{|x – 7| < -3}\)
6. \(\color{blue}{|4x – 8| > 0}\)

Answers

1. −6 < x < 6
2. x < −7 or x > 3
3. −\(\color{blue}{7 \le x \le 7}\)  (\(\color{blue}{\text{ from } -12 \le 3x-9 \le 12}\) → −\(\color{blue}{3 \le 3x \le 21}\) → −\(\color{blue}{1 \le x \le 7}\))
4. x ≤ −7 or \(\color{blue}{x \ge 3}\)
5. No solution (right side is negative)
6. All real numbers except \(\color{blue}{x = 2}\) (\(\color{blue}{4x-8}\) ≠ 0 → x ≠ 2)

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Free Solving Absolute Value Inequalities Worksheet

Ready to practice on your own? Download our free Solving Absolute Value Inequalities worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Solving Absolute Value Inequalities before a quiz or test.

Download Absolute Value Inequalities Worksheet

Frequently Asked Questions

How do I remember whether to use AND or OR?

Use this memory trick: “less-AND” and “great-OR.” Less-than absolute value inequalities give a compound AND (one interval); greater-than absolute value inequalities give a compound OR (two rays pointing outward).

What does the solution look like on a number line?

For a “less than” inequality, the solution is a bounded segment between two points (shaded between). For a “greater than” inequality, the solution is two rays pointing in opposite directions (shaded outside the points). Open dots are used for < or >; closed dots for ≤ or ≥.

Can I check my solution to an absolute value inequality?

Yes. Pick a value inside your solution set and one outside it, and substitute both back into the original inequality. The first should make it true; the second should make it false.

Related Topics

• Solving Absolute Value Equations
• Graphing Absolute Value Inequalities
• Graphing Absolute Value Equations
• Solving Inequalities
• Integers and Absolute Value