How to Solve Absolute Value Equations?

The absolute value of a number is its distance from zero on the number line, so it is always non-negative. When you are solving absolute value equations, you must account for both the positive and negative cases. This guide explains the method step by step, covers special cases (no solution, one solution), and includes worked examples, two video lessons, and practice problems.

What Is an Absolute Value Equation?

An absolute value equation is any equation that contains an absolute value expression such as \(\color{blue}{|\text{ expression }| = c}\). Because absolute value measures distance, the expression inside the bars can equal either \(\color{blue}{c}\) or \(\color{blue}{-c}\) — both are the same distance from zero.

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For example, \(\color{blue}{|x| = 5}\) means \(\color{blue}{x = 5}\) or x = −5, since both numbers are 5 units from zero.

How to Solve Absolute Value Equations

The Two-Case Rule

When the absolute value is isolated and c > 0, split into two equations:

|expression| = c ⇒ \(\color{blue}{\text{ expression } = c}\)   OR   expression = −c

Solve each equation separately and check both answers.

Special Case: No Solution

If the absolute value equals a negative number, there is no solution — absolute value can never be negative.

Quick example: \(\color{blue}{|3x + 1| = -5}\) → No solution

Special Case: One Solution

If the absolute value equals zero, there is exactly one solution.

Quick example: \(\color{blue}{|x – 4| = 0}\) → \(\color{blue}{x – 4 = 0}\) → \(\color{blue}{x = 4}\)

Step-by-Step Summary

1. Isolate the absolute value expression on one side.
2. Check the right side: if it is negative, write “no solution.” If it is zero, solve the single equation.
3. If the right side is positive, set up two cases: \(\color{blue}{\text{ expression } = c}\) and expression = −c.
4. Solve each case for the variable.
5. Check both solutions in the original equation.

Watch: Absolute Value Equations – Introduction (Video Lesson)

Math with Mr. J introduces absolute value equations and why two cases are needed:

Solving Absolute Value Equations – Worked Examples

Example 1: Solve \(\color{blue}{|2x – 3| = 7}\)

The absolute value is already isolated.
Case 1: \(\color{blue}{2x – 3 = 7}\) → \(\color{blue}{2x = 10}\) → x = 5
Case 2: \(\color{blue}{2x – 3}\) = −7 → 2x = −4 → x = −2
Check: |\(\color{blue}{2(5)-3}\)| = |7| = 7 ✓   |\(\color{blue}{2(-2)-3}\)| = |−7| = 7 ✓

Example 2: Solve \(\color{blue}{|x + 5| = 12}\)

Case 1: \(\color{blue}{x + 5 = 12}\) → x = 7
Case 2: \(\color{blue}{x + 5}\) = −12 → x = −17

Example 3: Solve \(\color{blue}{|3x + 1| = 10}\)

Case 1: \(\color{blue}{3x + 1 = 10}\) → \(\color{blue}{3x = 9}\) → x = 3
Case 2: \(\color{blue}{3x + 1}\) = −10 → 3x = −11 → x = −\(\color{blue}{\frac{11}{3}}\)
Check: |\(\color{blue}{3(3)+1}\)| = |10| = 10 ✓   |\(\color{blue}{3(-\frac{11}{3})+1}\)| = |−10| = 10 ✓

Example 4: Solve \(\color{blue}{|2x + 6| = -1}\)

The right side is negative. Absolute value can never equal a negative number.
No solution

Step-by-Step Solving Practice (Video Lesson)

Math with Mr. J solves a variety of absolute value equations with full step-by-step explanations:

Exercises: Solving Absolute Value Equations

1. \(\color{blue}{|x + 3| = 10}\)
2. \(\color{blue}{|4x – 8| = 12}\)
3. \(\color{blue}{|5x + 10| = 0}\)
4. \(\color{blue}{|3x – 9| = -6}\)
5. \(\color{blue}{|\frac{x}{2} + 1| = 4}\)
6. \(\color{blue}{|2x + 7| = 15}\)

Answers

1. \(\color{blue}{x = 7}\) or x = −13
2. \(\color{blue}{4x-8=12}\) → \(\color{blue}{x=5}\); \(\color{blue}{4x-8}\)=−12 → x=−1; \(\color{blue}{x = 5}\) or x = −1
3. \(\color{blue}{5x+10=0}\) → x=−2; x = −2 (one solution)
4. No solution (right side is negative)
5. \(\color{blue}{\frac{x}{2}+1=4}\) → \(\color{blue}{x=6}\); \(\color{blue}{\frac{x}{2}+1}\)=−4 → x=−10; \(\color{blue}{x = 6}\) or x = −10
6. \(\color{blue}{2x+7=15}\) → \(\color{blue}{x=4}\); \(\color{blue}{2x+7}\)=−15 → x=−11; \(\color{blue}{x = 4}\) or x = −11

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Free Solving Absolute Value Equations Worksheet

Ready to practice on your own? Download our free Solving Absolute Value Equations 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 Equations before a quiz or test.

Download Absolute Value Equations Worksheet

Frequently Asked Questions

Why do absolute value equations have two solutions?

Because absolute value measures distance from zero, both a positive and a negative number can have the same absolute value. For example, both 7 \(\color{blue}{\text{ and } -7}\) are 7 units from zero, so |x| = 7 has two solutions: \(\color{blue}{x = 7}\) and x = −7.

What does “no solution” mean for an absolute value equation?

It means there is no real number that satisfies the equation. This happens when the absolute value is set equal to a negative number, like |\(\color{blue}{x + 3}\)| = −5, because no real number has a negative distance from zero.

Do I always need to check my solutions?

Yes, especially when you manipulate the equation before isolating the absolute value (for example, multiplying both sides by a variable). Checking by substituting into the original equation confirms you have not introduced extraneous solutions.

Related Topics

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