Inequality Calculator
Solve linear, quadratic, and absolute-value inequalities and get interval notation, set-builder notation, boundary values, and checked steps.
∞ always uses a parenthesis, and ∪ joins separate solution intervals.Formulas used
For a linear inequality, isolate x by subtracting the constant and dividing by its coefficient. Dividing by a negative number reverses the comparison. For a quadratic inequality, the calculator finds the real roots with the discriminant D=b²−4ac and uses the parabola’s sign between and outside those roots. For |ax+b|<c, values lie between the two boundaries; for |ax+b|>c, they lie outside them.
Checked examples
| Inequality | Solution | Interval notation |
|---|---|---|
| 2x + 3 < 7 | x < 2 | (−∞, 2) |
| −3x + 6 ≥ 0 | x ≤ 2 | (−∞, 2] |
| x² − 5x + 6 ≤ 0 | 2 ≤ x ≤ 3 | [2, 3] |
| x² + 1 < 0 | No real solution | ∅ |
| −x² + 4 ≥ 0 | −2 ≤ x ≤ 2 | [−2, 2] |
| |2x − 4| < 6 | −1 < x < 5 | (−1, 5) |
| |x + 1| ≥ 3 | x ≤ −4 or x ≥ 2 | (−∞, −4] ∪ [2, ∞) |
How to use the calculator
- Choose linear, quadratic, or absolute-value mode.
- Select the comparison symbol.
- Enter the coefficients shown by the selected form.
- Calculate to see the solution in interval and set-builder notation.
- Review the boundaries and steps, especially when a negative linear coefficient reverses the sign.
Special cases and limits
- Zero leading coefficients are handled: a quadratic can reduce to a linear inequality, and a constant inequality can be true for every real number or none.
- Quadratic roots that are not integers are shown to about 10 significant digits.
- Each input must be finite and have absolute value no greater than 10¹².
- The calculator solves one real-variable inequality at a time; it does not parse typed algebraic expressions or chained input.
- Very large, nearly cancelling coefficients can be affected by normal floating-point rounding.
Frequently asked questions
When does an inequality sign reverse?
It reverses when both sides are multiplied or divided by a negative number. Adding or subtracting the same value does not reverse it.
What does an empty-set result mean?
∅ means no real x makes the inequality true. For example, x²+1<0 has no real solution.
Why can a quadratic answer have two intervals?
A parabola can be positive or negative outside its two roots and have the opposite sign between them. The union symbol joins the two separate ranges.
Are equality endpoints included?
Yes for ≤ and ≥, provided the boundary makes the original expression equal to the other side. They appear with brackets.
Related calculators
Use the Quadratic Formula Calculator to inspect roots and discriminants, the Polynomial Calculator for polynomial operations, the System of Equations Calculator for simultaneous equations, or the Number Line to visualize numeric positions.