What this calculator does
An absolute value equation usually has two solutions, because the expression inside can be either positive or negative and still give the same absolute value. Solving means splitting into those two cases.
Three outcomes are possible. A positive right-hand side gives two solutions, a value of zero gives exactly one, and a negative value gives none at all, since an absolute value can never be negative.
The formula
The equation splits into ax + b equals c and ax + b equals minus c. Each is solved for x, and both are checked by substituting back.
| Term | Meaning |
|---|---|
| Absolute value | Distance from zero, always non-negative. |
| Two cases | The expression inside can be positive or negative, giving two equations. |
| No solution | What results when the right-hand side is negative. |
The inputs explained
| Field | What to enter |
|---|---|
| a | Coefficient of x inside the absolute value. Cannot be zero. |
| b | Constant inside the absolute value. |
| c | The value on the right. Negative values have no solution. |
When to use it
Solving algebra problems
Absolute value equations are a standard topic and the two-case split is where errors happen.
Expressing a tolerance
A measurement within a tolerance is naturally an absolute value statement.
Understanding why there are two answers
Both solutions sit equally far from the point where the inside expression is zero.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the solutions spread?
The same expression at three right-hand values.
Questions
Why are there usually two solutions?
Because absolute value discards the sign. Both a positive and a negative value of the inside expression give the same absolute value, so two different x values satisfy the equation.
When is there only one solution?
When the right-hand side is exactly zero. The only way for an absolute value to be zero is for the expression inside to be zero, which happens at a single point.
Why can there be no solution?
Because an absolute value measures distance from zero and cannot be negative. Setting it equal to a negative number asks for something impossible.
How do absolute value inequalities differ?
They give intervals rather than points. Less than gives the region between the two solutions, and greater than gives the two regions outside them, which is a common source of confusion.
For quadratic equations, see the quadratic calculator. For solving exponentials, see the exponential equation calculator.