How to Solve Absolute Value Equations: Step-by-Step Guide

To solve an absolute value equation, first isolate the absolute value, then split it into two cases: |A| = b becomes A = b or A = −b. For example, |x − 3| = 5 gives x − 3 = 5 or x − 3 = −5, so x = 8 or x = −2. If the absolute value equals a negative number, there is no solution.

Absolute value equations look intimidating because the symbol is unfamiliar and the answer almost always splits into two cases. Once you see why the split happens, the topic becomes one of the most predictable in Algebra 1. Almost every absolute value problem on every quiz follows the same three-step recipe.

This guide walks through the recipe, every special case, the related inequality rules, and the five mistakes that cost most students points.

What Absolute Value Actually Means

|x| is the distance between x and 0 on the number line. Distance is always non-negative, so:

• |5| = 5.
• |−5| = 5.
• |0| = 0.
• |x| is never negative for any real x.

This single fact controls every problem.

The Two-Case Rule

If |expression| = a (where a ≥ 0), then either:

• expression = a, or
• expression = −a.

You solve both and report both answers (unless one is rejected, which we will get to).

Example. Solve |x| = 7. – Case 1: x = 7. – Case 2: x = −7.

Solution: x = ±7.

Solving Absolute Value Equations: The Three-Step Recipe

1. Isolate the absolute value expression. Get |…| alone on one side.
2. Split into two cases. Drop the bars and write two equations: one positive, one negated.
3. Solve each case and check.

Example 1: One Absolute Value

Solve |2x − 3| = 9.

Step 1: Already isolated. Step 2: Two cases. – 2x − 3 = 9 → 2x = 12 → x = 6. – 2x − 3 = −9 → 2x = −6 → x = −3.

Solutions: x = 6 or x = −3.

Example 2: Isolation First

Solve 3|x + 4| − 5 = 13.

Step 1: Add 5: 3|x + 4| = 18. Divide by 3: |x + 4| = 6. Step 2: Two cases. – x + 4 = 6 → x = 2. – x + 4 = −6 → x = −10.

Solutions: x = 2 or x = −10.

Example 3: No Solution

Solve |3x − 1| = −4.

Stop. Absolute value cannot equal a negative number. No solution.

This is the most-missed problem on every quiz. Look at the right side first; if it is negative, write “no solution” and move on.

Example 4: Single Solution

Solve |2x − 8| = 0.

Two cases collapse to one: – 2x − 8 = 0 → x = 4.

When the right side is 0, there is exactly one solution.

Absolute Value Equations With Two Absolute Values

If |A| = |B|, either A = B or A = −B. Same two-case rule, applied to both expressions.

Example. Solve |2x + 1| = |x − 4|.

Case 1: 2x + 1 = x − 4 → x = −5. Case 2: 2x + 1 = −(x − 4) → 2x + 1 = −x + 4 → 3x = 3 → x = 1.

Solutions: x = −5 or x = 1.

Check both, especially if the next step is an inequality.

Always Check Your Answers

Absolute value equations sometimes produce extraneous solutions when the right side contains a variable.

Example. Solve |x + 2| = 3x.

Case 1: x + 2 = 3x → 2 = 2x → x = 1. Check: |1 + 2| = 3 and 3(1) = 3. Valid. Case 2: x + 2 = −3x → 4x = −2 → x = −1/2. Check: |−1/2 + 2| = 3/2 but 3(−1/2) = −3/2. The right side is negative; rejected.

Final solution: x = 1.

When the right side has a variable, always plug both answers back in.

Absolute Value Inequalities: A Two-Rule Trick

The two rules every student should memorize:

• |expression| < a becomes −a < expression < a. (“Less than” means “and.”)
• |expression| > a becomes expression < −a OR expression > a. (“Greater than” means “or.”)

These rules also work with ≤ and ≥.

Inequality Example 1

Solve |2x − 5| ≤ 7.

Rewrite: −7 ≤ 2x − 5 ≤ 7. Add 5 across: −2 ≤ 2x ≤ 12. Divide by 2: −1 ≤ x ≤ 6.

Graph: closed circles on −1 and 6, shaded between.

Inequality Example 2

Solve |x − 3| > 4.

Rewrite: x − 3 < −4 OR x − 3 > 4. Solve each: x < −1 OR x > 7.

Graph: open circles on −1 and 7, shaded outward in both directions.

Special Inequality Cases

• |expression| < negative number: no solution.
• |expression| > negative number: all real numbers.
• |expression| < 0: no solution (since absolute value is never negative).
• |expression| ≥ 0: all real numbers (always true).

These special cases are favorite test items. Recognize the form, write the answer, save time.

Absolute Value Equations in Word Problems

Most word problems with absolute value describe a distance, a tolerance, or a margin of error.

A factory produces bolts. The acceptable diameter is 5 mm with a tolerance of 0.1 mm. Write and solve an absolute value inequality for acceptable diameters d.

Setup: |d − 5| ≤ 0.1. Rewrite: −0.1 ≤ d − 5 ≤ 0.1. Add 5: 4.9 ≤ d ≤ 5.1.

Acceptable diameters are 4.9 mm to 5.1 mm.

A thermostat keeps room temperature within 2 degrees of 70 °F. Express the range with an absolute value inequality.

|T − 70| ≤ 2, which gives 68 ≤ T ≤ 72.

Tolerance problems are the easiest absolute value word problems on standardized tests. Recognize the structure once and you collect points all year.

Common Mistakes

1. Forgetting to isolate first. Trying to split |2x − 3| + 5 = 12 without subtracting 5 first leads to garbage.
2. Skipping the negative case. A solution like |x| = 5 has two answers, not one.
3. Missing the “no solution” trap. |x − 2| = −3 has no solution. Always check the right side.
4. Confusing inequality direction. “Less than” means “and”; “greater than” means “or.” Write that on your scratch paper before the inequality unit test.
5. Skipping the check on variable right sides. Extraneous solutions are common when the right side contains x.

Quick Recap Cheat Sheet

Tape this cheat sheet inside the front cover of your binder.

Frequently Asked Questions

What is the first step in solving absolute value equations? The first step is to isolate the absolute value expression on one side before splitting into cases. For 2|x − 1| + 3 = 11, subtract 3 to get 2|x − 1| = 8, then divide by 2 to get |x − 1| = 4. Now split: x − 1 = 4 or x − 1 = −4, so x = 5 or x = −3.

What is no solution in absolute value equations? An absolute value equation has no solution when, after isolating it, the absolute value equals a negative number. For example, |x + 6| = −2 has no solution because absolute value is a distance and can never be negative. Isolate first, though: |x| − 4 = −1 looks negative but becomes |x| = 3, which has the solutions x = 3 and x = −3.

What is an extraneous solution in absolute value equations? An extraneous solution is a value you get from one of the two cases that does not work in the original equation. It often appears when a variable is outside the bars. For |x + 2| = 2x − 1, the cases give x = 3 and x = −1/3. Checking, x = 3 works (5 = 5), but x = −1/3 gives 5/3 = −5/3, so it is extraneous.

How do you solve absolute value equations with two absolute values? For |A| = |B|, the expressions inside are either equal or opposites, so solve A = B and A = −B. For |x − 1| = |2x + 4|: x − 1 = 2x + 4 gives x = −5, and x − 1 = −(2x + 4) gives x = −1. Both check: at x = −5 each side is 6, and at x = −1 each side is 2.

How do you solve absolute value equations and inequalities? Equations split into two equations: |A| = b means A = b or A = −b. Inequalities follow two rules: |A| < b becomes the compound inequality −b < A < b, and |A| > b becomes A < −b or A > b. For example, |x − 2| < 5 becomes −5 < x − 2 < 5, so −3 < x < 7.

How do you solve absolute value equations graphically? Graph each side of the equation as its own function and find where the graphs intersect; the x-coordinates of the intersection points are the solutions. For |x − 3| = 5, graph the V-shaped y = |x − 3| and the horizontal line y = 5. They cross at x = −2 and x = 8. If the graphs never meet, the equation has no solution.

Closing Thought

Absolute value equations and inequalities are a three-step recipe and a two-rule trick. Isolate, split, check. For inequalities, "less than" becomes "and"; "greater than" becomes "or." Lock those in and the topic becomes one of your easiest grade boosters in Algebra 1.

For more practice, browse our Algebra 1 worksheets and our full Math Topics library. When you are ready for a structured workbook, our Algebra 1 collection covers every type above.

Related: Exponents and Radicals