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).

Algebra I for Beginners 2026 The Ultimate Step by Step Guide to Acing Algebra I

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

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)

Pre-Algebra for Beginners 2026 The Ultimate Step by Step Guide to Preparing for the Pre-Algebra Test

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Rated 4.30 out of 5 based on 105 customer ratings

Satisfied 92 Students

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 you solve an absolute value inequality?

First isolate the absolute value. For a “less than” inequality, write a compound “and” inequality: |x − 3| < 5 becomes −5 < x − 3 < 5, so −2 < x < 8. For a “greater than” inequality, split it into two “or” inequalities. Finally, test a value from your answer in the original inequality.

When solving an absolute value inequality do you flip the sign?

You flip it for the negative case. For |2x + 1| ≥ 7, solve 2x + 1 ≥ 7 and 2x + 1 ≤ −7, where the sign flips as the 7 becomes −7. That gives x ≥ 3 or x ≤ −4. You also flip the sign any time you multiply or divide both sides by a negative number.

What is an absolute value inequality that has no solution?

An inequality like |x + 4| < −2 has no solution, because an absolute value is never negative and can never be less than a negative number. The same is true for |x| ≤ −1. Before solving, check the number on the other side once the absolute value is isolated.

Can you solve an absolute value inequality with a negative number?

Yes, but the answer is either no solution or all real numbers. If the absolute value is less than a negative number, as in |x − 1| < −3, nothing works. If it is greater than a negative number, as in |x − 1| > −3, every real number works, since an absolute value is always 0 or more.

How do you solve absolute value inequalities in interval notation?

Solve as usual, then write the answer as intervals. |x + 2| ≤ 3 gives −5 ≤ x ≤ 1, which is [−5, 1]. The opposite inequality, |x + 2| > 3, gives x < −5 or x > 1, written (−∞, −5) ∪ (1, ∞). Use brackets for ≤ or ≥ and parentheses for < or >.

How do you solve an absolute value inequality word problem?

Write the distance from a target value as an absolute value. If a room should stay within 3 degrees of 70°F, write |t − 70| ≤ 3. Solve it as −3 ≤ t − 70 ≤ 3, so 67 ≤ t ≤ 73. The answer means any temperature from 67°F to 73°F is acceptable.

Related Topics

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