“Less than” absolute value inequalities describe numbers close to a point and give a single interval; “greater than” inequalities describe numbers far from a point and give two separate rays. Tolerances in engineering (“within ±0.2 mm”) are classic examples.
How to use the Absolute Value Inequalities Calculator
- a (coefficient of x)
- b (constant)
- Inequality sign
- 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
The calculator rewrites the inequality as a compound inequality, solves for x and adjusts for a negative a (which flips the inequality signs). Strict inequalities give open brackets, non-strict ones closed brackets.
Formula
|u| < c ⇔ −c < u < c
|u| > c ⇔ u < −c or u > c
Worked example
Inputs — a (coefficient of x): 2; b (constant): -1; Inequality sign: < (less than); c (right-hand side): 5.
Results — Solution set (interval notation) = (−2, 3); Rewritten without absolute values = |2x − 1| < 5 → −5 < 2x − 1 < 5; Boundary points = Boundary points: x = −2 and x = 3.
Key terms
- Compound inequality
- Two inequalities joined by "and" or "or".
- Interval notation
- Writing a set of numbers as (a, b), [a, b] and so on.
Frequently asked questions
What is the solution of
x| < 3? | All x between −3 and 3: (−3, 3).
What is the solution of
x| > 3? | x < −3 or x > 3: (−∞, −3) ∪ (3, ∞).
What if c is negative?
|ax + b| < c has no solution; |ax + b| > c is true for every x.
What do round and square brackets mean?
A round bracket excludes the endpoint, a square bracket includes it.