Math
Absolute Value Equation Calculator
Choose the form |ax + b| = c or |ax + b| = |cx + d|, enter the numbers, and the absolute value equation calculator splits the problem into its positive and negative cases, solves each exactly, checks the answers by substituting back, and explains when there is no solution or every number works.
Use 1 for |x + b|.
For |ax + b| = c this is the right-hand side. For the two-absolute-value form it is the coefficient of x inside the second absolute value.
Try an example
Result
Solution
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- Equation
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- Number of solutions
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- Solutions as decimals
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- Case 1
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- Case 2
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- Distance interpretation
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More details (1 more)
- Why
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the absolute value equation calculator solves
How far a number sits from zero, so the answer is never negative. measures distance from 0, so |u| = c means u is c units from 0 in either direction. That is why an absolute value equation usually has two answers: one where the inside is positive and one where it is negative. When two absolute values are equal, the insides are either equal or opposites.
How many solutions to expect
| Situation | Solutions | Why |
|---|---|---|
| |ax + b| = c with c > 0 | 2 | The inside can be c or -c |
| |ax + b| = 0 | 1 | Only 0 has absolute value 0 |
| |ax + b| = c with c < 0 | 0 | Absolute value is never negative |
| |ax + b| = |cx + d|, lines not parallel | 2 (or 1 if the cases agree) | Equal or opposite insides |
| |ax + b| = |cx + d| with identical or opposite insides | All real numbers | One case is always true |
How to use it
- Pick the form. For |ax + b| = c enter a, b, and the right-hand side c. For |ax + b| = |cx + d| enter all four numbers.
- Read the solutions at the top. Each case and its answer is listed, followed by a substitution check.
- If the right side is negative the calculator reports no solution and explains why.
How to read the answer
Each solution makes the inside of the absolute value equal to the right side or its opposite. On a number line, the two solutions of |x - h| = c sit c units to the left and right of h.
Common mistakes and edge cases
- Forgetting the negative case and reporting only one answer.
- Trying to solve |ax + b| = c with c negative. There is no solution, no matter what a and b are.
- Not isolating the absolute value before splitting into cases.
- Distributing the minus sign to only one term in -(cx + d).
Worked examples
Two solutions
|2x - 3| = 7 gives x = -2 or x = 5
Solution
x = -2 or x = 5
Fraction answers
|3x + 1| = 4 gives x = -5/3 or x = 1
Solution
x = -5/3 or x = 1
One solution
|4x - 8| = 0 gives x = 2
Solution
x = 2
No solution
|x + 1| = -4 has no solution
Solution
No solution
Two absolute values
|2x + 1| = |x - 3| gives x = -4 or x = 2/3
Solution
x = -4 or x = 2/3
All real numbers
|x + 2| = |-x - 2| is always true
Solution
All real numbers
Decimal coefficients
|0.5x - 1| = 2.5 gives x = -3 or x = 7
Solution
x = -3 or x = 7
Frequently asked questions
Why do absolute value equations have two solutions?+
Because |u| = c is true when u = c and when u = -c: both numbers are c units from zero. Each choice gives one linear equation to solve.
When does an absolute value equation have no solution?+
When the isolated absolute value equals a negative number, such as |x + 1| = -4. An absolute value can never be negative, so nothing works.
How do I solve |ax + b| = |cx + d|?+
Set the insides equal (ax + b = cx + d) and set them opposite (ax + b = -(cx + d)). Solve both linear equations. Usually you get two answers; if the insides are identical or exact opposites, every real number works.
Do I need to check the answers?+
For these linear absolute value equations the two cases never create extraneous answers, but checking by substitution is still the fastest way to catch arithmetic mistakes. The calculator shows the check for every solution.
What if the coefficient of x is 0?+
Then the variable disappears. |b| = c is either always true (every real number is a solution) or always false (no solution), and the calculator says which.
What does the distance interpretation mean?+
|ax + b| = c can be rewritten as |x - h| = c/|a| with h = -b/a, which says x is c/|a| units away from h on the number line.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026