Absolute value inequality solver
Pick the sign and enter a, b and c to get the full solution set, including the special cases when c is zero or negative.
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How to use it
- Enter coefficient a, constant b, sign and right side c.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
|ax + b| < c means -c < ax + b < c (between two values), and |ax + b| > c means ax + b < -c or ax + b > c (outside them). A negative c gives no solution for the first and all x for the second.
Example
Inputs: Coefficient a = 2, Constant b = 1, Sign = greater than or equal (≥), Right side c = 5. Result: Solution: x ≤ -3 or x ≥ 2.
Limits and caveats
- Covers |ax + b| compared with a constant c only.
- Endpoints are rounded to 10 significant digits.
Questions about this tool
When is the answer one interval and when two?
"Less than" gives one interval between two values; "greater than" gives two intervals outside them.
How do I solve |x - 5| ≤ 3?
Enter a = 1, b = -5, choose ≤ and c = 3. The answer is 2 ≤ x ≤ 8.
Why is |ax + b| > -2 true for every x?
An absolute value is never below zero, so it is always greater than any negative number.