Inequality calculator
Solve an inequality step by step and get the answer three ways: as an inequality, in interval notation, and on a number line.
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Solving inequalities: the rules
An inequality is solved almost exactly like an equation. You can add, subtract, multiply or divide both sides by the same number, with one exception:
Multiplying or dividing by a negative number reverses the sign. Start from and multiply by −1: . The order of the numbers flips on the number line, so the sign must flip too.
| Inequality | Interval | Number line |
|---|---|---|
| open dot at 2, arrow right | ||
| filled dot at 2, arrow left | ||
| open at −1, filled at 4, shaded between | ||
| two rays pointing outwards |
Quadratic and rational inequalities
These can change sign only where the expression is zero or undefined. The solver finds those critical points, splits the number line into intervals and tests one value in each, which is the sign-chart method taught in most Algebra 2 courses. Points that make a denominator zero are never included.
Frequently asked questions
When do I flip the inequality sign?
Only when you multiply or divide both sides by a negative number. −2x > 6 becomes x < −3. Adding or subtracting never flips it.
What is interval notation?
A compact way to write a solution set. Round brackets ( ) mean the endpoint is not included; square brackets [ ] mean it is. ∞ always gets a round bracket. So x ≥ 2 is [2, ∞), and −1 < x ≤ 4 is (−1, 4].
How do you solve a quadratic inequality?
Move everything to one side, find where the expression equals zero (the critical points), then test one value in each interval between them to see where the inequality holds.
How do absolute value inequalities work?
|u| < k means u is within k of zero, so −k < u < k. |u| > k means u is further than k from zero, so u < −k or u > k.