Solves an equation of the form |x − a| = b for both possible values of x.
How it works
An absolute value equation splits into two cases — the expression inside equals b, or it equals negative b — giving two solutions.
What this does not include
This does not include simply evaluating |x| for a known x — for that, use this site’s absolute value calculator instead.
How to use this calculator
- Enter the a and b values from your equation |x − a| = b.
A worked example
|x − 3| = 5: two solutions, x = 3+5 = 8 and x = 3−5 = −2.
|x + 2| = 4: solutions x = 2 and x = −6.
What the variables mean
| Variable | Meaning |
|---|---|
| a | The value being offset from inside the absolute value |
| b | The target absolute value on the right side of the equation |
Edge cases worth knowing
An absolute value equation almost always has two solutions, not one. Since |x−a| = b means x−a could be either +b or −b, both branches need solving — a common mistake is stopping after finding just one.
A negative b has no solution — an absolute value can never equal a negative number, so the calculator correctly declines to show one for that case.
Frequently asked questions
Why does an absolute value equation have two solutions?
Because both a positive and a negative value inside the absolute value bars can produce the same result once the sign is stripped away.
What if b is negative?
There’s no solution — an absolute value can never equal a negative number, since it always measures a non-negative distance.
What if b is zero?
Both solutions collapse into a single value, x = a, since the only way |x − a| can equal zero is if x − a itself is zero.
