What this calculator does
An absolute value inequality splits into two ordinary inequalities, and which way it splits depends entirely on the direction of the comparison. Less than gives a single interval between two boundaries, because the expression has to stay close to zero. Greater than gives two rays running outwards, because the expression has to be far from zero in either direction.
Getting that distinction the wrong way round is the standard error, and it is easy to make because the algebra looks almost identical either way. The other trap is a negative value on the right: an absolute value can never be less than a negative number, so those cases have no solution at all, while greater than a negative number is satisfied by every real number.
The formula
For a less-than comparison, |ax + b| < c becomes −c < ax + b < c, which is solved for x to give a single interval between the two boundary values. For a greater-than comparison, the solution is everything outside that interval instead, written as a union of two rays. The boundaries themselves come from solving ax + b = c and ax + b = −c, and whether they are included depends on whether the comparison is strict.
| Term | Meaning |
|---|---|
| |ax + b| | The absolute value of the expression, its distance from zero regardless of sign. |
| Interval notation | Round brackets for a boundary that is excluded, square brackets for one that is included. |
| Union | The ∪ symbol joining two separate ranges, which is how a greater-than solution is written. |
| Boundary value | A value of x where the expression inside the bars equals exactly c or −c, marking where the solution starts or stops. |
The inputs explained
| Field | What to enter |
|---|---|
| a (coefficient of x) | The coefficient of x inside the absolute value bars. It cannot be zero, since there would then be no x to solve for. |
| b (constant inside) | The constant term inside the bars, added to ax before the absolute value is taken. |
| Compared to | The direction of the comparison, and whether the boundary values themselves are included. |
| c | The value on the right-hand side. A negative value here gives either no solution or all real numbers, depending on the direction chosen. |
When to use it
Expressing a tolerance band
A specification such as a measurement within 0.5 mm of a target is exactly an absolute value inequality, and solving it gives the acceptable range in the form a person can actually check against.
Working through an algebra exercise
Absolute value inequalities appear in every introductory algebra course, and the interval notation is often marked as strictly as the arithmetic. Having both the intervals and the boundary values makes it easy to check each part separately.
Checking the degenerate cases
Negative and zero values of c produce answers that are not intervals at all, and those are precisely the cases an exercise is likely to test. Running them here confirms what the answer should look like.
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 does the direction of the comparison change the solution?
The same expression and the same right-hand side, compared four different ways.
| Comparison | Solution set | Boundary values |
|---|---|---|
| < 5 | (-1.000, 4.000) | x = -1.000 and x = 4.000 |
| ≤ 5 | [-1.000, 4.000] | x = -1.000 and x = 4.000 |
| > 5 | (−∞, -1.000) ∪ (4.000, ∞) | x = -1.000 and x = 4.000 |
| ≥ 5 | (−∞, -1.000] ∪ [4.000, ∞) | x = -1.000 and x = 4.000 |
What happens as the right-hand side shrinks to zero and below?
A less-than comparison with the right-hand side reduced step by step, into the degenerate cases.
| c | Solution set | Boundary values |
|---|---|---|
| 10 | (-3.500, 6.500) | x = -3.500 and x = 6.500 |
| 5 | (-1.000, 4.000) | x = -1.000 and x = 4.000 |
| 1 | (1.000, 2.000) | x = 1.000 and x = 2.000 |
| 0 | No solution | n/a |
| -2 | No solution (an absolute value can never be less than a negative number) | n/a |
Questions
Why does a greater-than inequality give two intervals instead of one?
Because the expression has to be far from zero, and there are two ways to be far from zero. Anything above c works and anything below −c works, but the range between them does not, so the solution is the two outer pieces joined by a union.
What happens when c is negative?
An absolute value is never negative, so it can never be less than a negative number and the less-than cases have no solution. The greater-than cases are satisfied by every real number, since any absolute value already exceeds any negative one.
When should a bracket be round rather than square?
Round when the boundary is excluded, which is the strict comparisons < and >. Square when it is included, which is ≤ and ≥. Infinity always takes a round bracket, because it is not a value that can be reached.
Does a negative coefficient a change the answer?
Not the solution set, though it swaps which boundary is which during the working. Dividing an inequality by a negative number reverses its direction, and doing that to both halves at once puts the smaller boundary back on the left, so the final interval reads the same way round.
For the equation version without the inequality, see the absolute value equation calculator. For a plain linear inequality with no absolute value involved, see the greater than or less than calculator.