Inequality Calculator
Solve linear, quadratic, polynomial, rational and modulus inequalities, such as 2x − 3 < 5x + 6, (x − 1)/(x + 2) ≥ 0 or |2x − 3| < 5, with a sign chart, a number line and interval notation.
About the Inequality Calculator
Solve an inequality like an equation, with one extra rule: multiplying or dividing both sides by a negative number reverses the sign, so < becomes > and ≤ becomes ≥.
The answer is a range of values. On the number line an open circle means the boundary isn't included (< or >) and a filled circle means it is (≤ or ≥).
For x², higher powers, fractions and |…|, the sign can only change at critical points. Mark them on the line, test one value in each stretch, and keep the stretches with the right sign.
The formula
If a < b and k < 0, then ka > kb
- (−3, ∞)
- every number greater than −3
- [2, ∞)
- 2 and every number greater than it
Worked example
2x − 3 < 5x + 6
- Subtract 5x from both sides: −3x − 3 < 6
- Add 3 to both sides: −3x < 9
- Divide by −3 and reverse the sign: x > −3
Questions people ask
Why does the sign flip with a negative?
Multiplying by −1 reverses order: 2 < 3, but −2 > −3.
Can it solve x² and fraction inequalities?
Yes. It moves everything to one side as one fraction, finds the critical points (zeros of the top, and zeros of the bottom, which are always excluded) and tests the sign in each stretch (the wavy-curve method). Never multiply both sides by an expression in x whose sign you don't know.
How are |…| inequalities solved?
Split at the point where the inside is 0: there |u| is u on one side and −u on the other. Solve each piece and join the answers. |x − a| < k means a − k < x < a + k.
What about 1 < x < 5?
Solve each part separately: 1 < x and x < 5, then combine them.