An absolute value equation has the form |ax + b| = c. Because the absolute value of any expression is always non-negative, the equation can have zero, one, or two real solutions depending on the value of c. This calculator finds all solutions and shows every step.
How to Use This Calculator
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Enter the coefficient a of x inside the absolute value bars. Use 1 if there is no explicit coefficient.
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Enter the constant b added inside the absolute value. Use 0 if there is none.
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Enter the right-hand side c. If c is negative, the equation has no real solution.
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Read the solutions and the step-by-step working below.
How to Solve Absolute Value Equations
The key property of absolute value is |expression| = k means expression = k or expression = −k (when k > 0). This splits one equation into two linear equations.
When k = 0, both cases give the same equation (expression = 0), so there is exactly one solution.
When k < 0, there is no solution because an absolute value can never be negative.
Solution Method
For the equation |ax + b| = c:
If c < 0: no real solution
If c = 0: ax + b = 0, so x = −b / a (one solution)
If c > 0: ax + b = c → x₁ = (c − b) / a, and ax + b = −c → x₂ = (−c − b) / a
Worked Example: |x − 3| = 5
Here a = 1, b = −3, c = 5. Since c > 0, there are two solutions.
Case 1: x − 3 = 5 → x = 8
Case 2: x − 3 = −5 → x = −2
Check x = 8: |8 − 3| = |5| = 5 ✓
Check x = −2: |−2 − 3| = |−5| = 5 ✓
The solution set is {−2, 8}.
Worked Example: |2x + 1| = 7
a = 2, b = 1, c = 7. Two solutions.
Case 1: 2x + 1 = 7 → 2x = 6 → x = 3
Case 2: 2x + 1 = −7 → 2x = −8 → x = −4
Check x = 3: |2(3) + 1| = |7| = 7 ✓
Check x = −4: |2(−4) + 1| = |−7| = 7 ✓
The solution set is {−4, 3}.
Solution Cases Summary
| Condition | Number of solutions | Example |
|---|---|---|
| c < 0 | No solution | |x + 2| = −3 → none |
| c = 0 | One solution | |x + 4| = 0 → x = −4 |
| c > 0, gives two distinct values | Two solutions | |x − 3| = 5 → x = 8 or x = −2 |
| c > 0, a = 0 and |b| = c | All real x (identity) | |0·x + 5| = 5 → always true |
| c > 0, a = 0 and |b| ≠ c | No solution | |0·x + 3| = 5 → never true |
Absolute Value on a Number Line
|x − h| = r describes all points whose distance from h on the number line equals r. The two solutions x = h + r and x = h − r are symmetric about h.
For |x − 3| = 5, the center is h = 3 and the radius is r = 5. The solutions x = 8 and x = −2 are each exactly 5 units from 3.
This geometric interpretation explains why c < 0 has no solution (negative distance is meaningless) and c = 0 has exactly one solution (both solutions collapse to the center point).
This calculator solves absolute value equations of the form |ax + b| = c. Enter the coefficients and the right-hand side to get all solutions instantly, with a full breakdown of each step.