How to Solve Absolute Value Equations?

Solve absolute value equations by using distance from zero and checking the resulting cases.

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.

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.

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

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 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

What is the first step in solving absolute value equations?

The first step is to isolate the absolute value, getting |expression| alone on one side. In 2|x − 3| + 1 = 9, subtract 1 to get 2|x − 3| = 8, then divide by 2 to get |x − 3| = 4. Only after that do you split into two cases. Splitting too early is one of the most common mistakes.

How do you solve absolute value equations step by step?

Isolate the absolute value, check that the other side is not negative, then write two equations: one with the positive value and one with the negative. Solve both and check each answer. For |x − 3| = 4, write x − 3 = 4 and x − 3 = −4, giving x = 7 or x = −1. Both check, since |4| = 4 and |−4| = 4.

What is no solution in absolute value equations?

An absolute value equation has no solution when the isolated absolute value equals a negative number, because a distance from zero can never be negative. For example, |x + 5| = −3 has no solution. Be careful to isolate first: |x| + 7 = 2 becomes |x| = −5, so it also has no solution, even though the 2 looked positive.

What is an extraneous solution in absolute value equations?

An extraneous solution is an answer you get from the algebra that does not work in the original equation. This happens when a variable is outside the absolute value. In |x + 2| = 3x, the cases give x = 1 and x = −1/2. Checking x = −1/2 gives |1.5| = 1.5 on the left but −1.5 on the right, so only x = 1 is a real solution.

How do you solve absolute value equations on both sides?

When both sides are absolute values, like |x − 1| = |2x + 3|, the insides are either equal or opposites. Case 1: x − 1 = 2x + 3 gives x = −4. Case 2: x − 1 = −(2x + 3) gives 3x = −2, so x = −2/3. Check both: |−5| = |−5| and |−5/3| = |5/3|, so both work.

How do you solve absolute value equations graphically?

Graph each side as its own function and find where the graphs meet. For |x − 2| = 3, graph the V-shaped y = |x − 2| and the horizontal line y = 3. They cross at x = −1 and x = 5, which are the solutions. If the line sits below the V’s vertex, the graphs never meet and there is no solution.

Related Topics

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