An absolute value measures distance from zero, so |ax + b| = c asks which x values put ax + b exactly c units from zero. There are usually two answers, one on each side, which is why absolute value equations split into two ordinary linear equations.
How to use the Absolute Value Equation Calculator
- a (coefficient of x)
- b (constant)
- c (right-hand side)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
If c is negative there is no solution, because a distance cannot be negative. If c = 0 there is a single solution x = −b/a. Otherwise the calculator solves ax + b = c and ax + b = −c separately.
Tips
- Rewrite the equation in standard form before reading off the coefficients.
- Check your answer by substituting it back into the original equation.
- Keep exact forms (fractions and square roots) until the final step to avoid rounding errors.
- For inequalities, flip the sign whenever you multiply or divide by a negative number.
Common mistakes to avoid
- Dropping a negative sign when copying a coefficient.
- Forgetting the ± when taking a square root.
- Dividing by an expression that could be zero.
Formula
|ax + b| = c ⇔ ax + b = c or ax + b = −c (c ≥ 0)
Worked example
Inputs — a (coefficient of x): 2; b (constant): -1; c (right-hand side): 5.
Results — Solution = x = −2 or x = 3; Split into two equations = |2x − 1| = 5 → 2x − 1 = 5 or 2x − 1 = −5; Boundary points = Boundary points: x = −2 and x = 3.
Key terms
- Absolute value
- The distance of a number from zero, |x|.
Frequently asked questions
Why are there two solutions?
Two numbers have the same distance from zero – for example 5 and −5 – so ax + b can equal either c or −c.
When is there no solution?
When c is negative. |anything| is never negative.
Can an absolute value equation have one solution?
Yes, when c = 0: the only solution is x = −b/a.
How do I check the answers?
Substitute each x into |ax + b| and confirm you get c.
What formula does the Absolute Value Equation Calculator use?
|ax + b| = c ⇔ ax + b = c or ax + b = −c (c ≥ 0).