Algebra
How to Solve Absolute Value Equations: Worked Examples
Isolate the absolute value, split into two cases, and check for extraneous solutions with clear step-by-step examples
By AI Picture Answer Education Team · · Updated September 25, 2026
What is absolute value?
The absolute value of a number is its distance from zero on the number line. Distance is never negative, so |5| = 5 and |−5| = 5. That single idea explains how to solve absolute value equations: if |x| = 5, then x is 5 units from zero, which means x can be 5 or −5. Most absolute value equations have two solutions for exactly this reason.
To check your answers as you practice, try the absolute value calculator, which solves |ax + b| = c problems and shows both cases.
Quick rule:
If |A| = c and c > 0, then A = c or A = −c. If c = 0, then A = 0. If c < 0, there is no solution.
How to solve an absolute value equation
- Isolate the absolute value so the equation looks like |expression| = number. Undo any addition, subtraction, or multiplication outside the bars first.
- Check the right side. If it is negative, stop: there is no solution. If it is zero, set the inside equal to zero.
- Split into two equations: expression = number, and expression = −(number).
- Solve both equations separately.
- Check each answer in the original equation. This step is required when a variable appears outside the absolute value bars.
Worked examples
Example 1: A basic two-case equation
Solve |2x − 3| = 7.
Step 1: The absolute value is already isolated, and 7 is positive, so there are two cases.
Case 1: 2x − 3 = 7. Add 3: 2x = 10. Divide by 2: x = 5.
Case 2: 2x − 3 = −7. Add 3: 2x = −4. Divide by 2: x = −2.
Answer: x = 5 or x = −2
Check: x = 5 gives |10 − 3| = |7| = 7. x = −2 gives |−4 − 3| = |−7| = 7.
Example 2: Isolate the absolute value first
Solve 3|x + 4| − 5 = 10.
Step 1: Add 5 to both sides: 3|x + 4| = 15.
Step 2: Divide both sides by 3: |x + 4| = 5.
Case 1: x + 4 = 5, so x = 1.
Case 2: x + 4 = −5, so x = −9.
Answer: x = 1 or x = −9
Check: x = 1 gives 3|5| − 5 = 15 − 5 = 10. x = −9 gives 3|−5| − 5 = 15 − 5 = 10.
Example 3: No solution
Solve |x − 2| + 6 = 1.
Step 1: Subtract 6 from both sides: |x − 2| = −5.
Step 2: An absolute value is a distance, so it can never equal a negative number.
Answer: No solution (∅)
Note: If you had split into cases anyway, you would get x = −3 and x = 7. Plugging in x = 7 gives |5| + 6 = 11, not 1, which confirms both are false.
Example 4: A variable on both sides (extraneous solution)
Solve |x + 1| = 2x − 4.
Case 1: x + 1 = 2x − 4. Subtract x from both sides: 1 = x − 4. Add 4: x = 5.
Case 2: x + 1 = −(2x − 4) = −2x + 4. Add 2x: 3x + 1 = 4. Subtract 1: 3x = 3. So x = 1.
Check x = 5: Left side |5 + 1| = 6. Right side 2(5) − 4 = 6. Valid.
Check x = 1: Left side |1 + 1| = 2. Right side 2(1) − 4 = −2. 2 ≠ −2, so x = 1 is extraneous.
Answer: x = 5
Example 5: Absolute value on both sides
Solve |x − 3| = |2x + 1|.
Idea: Two absolute values are equal when the insides are equal or opposites.
Case 1: x − 3 = 2x + 1. Subtract x: −3 = x + 1. Subtract 1: x = −4.
Case 2: x − 3 = −(2x + 1) = −2x − 1. Add 2x: 3x − 3 = −1. Add 3: 3x = 2. So x = 2/3.
Answer: x = −4 or x = 2/3
Check: x = −4 gives |−7| = 7 and |−8 + 1| = |−7| = 7. x = 2/3 gives |2/3 − 3| = |−7/3| = 7/3 and |4/3 + 1| = |7/3| = 7/3.
Special case: Equal to zero
Solve |5x − 10| = 0. Only zero has an absolute value of zero, so there is just one case: 5x − 10 = 0, which gives x = 2. Splitting into two cases here would give the same answer twice.
What about absolute value inequalities?
The same distance idea extends to inequalities. |x| < 4 means x is less than 4 units from zero, so −4 < x < 4. |x| > 4 means x is more than 4 units from zero, so x < −4 or x > 4. These become compound inequalities, which we cover in detail in how to solve compound inequalities.
Common mistakes to avoid
Watch out for these:
- Splitting before isolating. In 3|x + 4| − 5 = 10, writing 3(x + 4) − 5 = ±10 gives x = 1 and x = −17/3, and the second value is wrong. Isolate |x + 4| first.
- Moving numbers into the bars. |x − 2| + 6 is not the same as |x + 4|. Terms outside the bars must be moved to the other side, never combined with what is inside.
- Forgetting the negative case. Most equations have two solutions. Writing only x = 5 for |2x − 3| = 7 misses x = −2.
- Negating only part of the right side. In Case 2, the whole side is negated: −(2x − 4) = −2x + 4, not −2x − 4.
- Skipping the check. When a variable appears outside the bars, one of your answers may be extraneous.
Practice problems
Try these on your own:
- Solve |x + 6| = 2.
- Solve 2|3x − 1| + 4 = 14.
- Solve |4 − x| = −3.
Click to see answers
- x + 6 = 2 or x + 6 = −2, so x = −4 or x = −8
- |3x − 1| = 5, so 3x − 1 = 5 (x = 2) or 3x − 1 = −5 (x = −4/3)
- No solution, because an absolute value cannot be negative
Frequently asked questions
Why do absolute value equations usually have two solutions?
Absolute value measures distance from zero, and two numbers (one positive, one negative) sit the same distance from zero. Each one gives a separate solution.
When does an absolute value equation have no solution?
When the isolated absolute value is set equal to a negative number, such as |x − 2| = −5, there is no solution.
What is an extraneous solution?
It is a value that comes out of the algebra but does not satisfy the original equation. It often appears when a variable is outside the absolute value bars.
Need help with absolute value equations?
Absolute value equations connect to inequalities and interval notation, so pair this guide with how to write interval notation. When you want to check a homework answer, enter it into the absolute value calculator or the compound inequality calculator.
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