Inequalities

The inequality solver solves linear inequalities ax + b ⋚ 0, quadratic inequalities ax² + bx + c ⋚ 0 and absolute value inequalities |ax + b| ⋚ c. The answer is given in interval notation with a step-by-step solution.

Linear inequalities

They are solved like equations, with one important rule: when you multiply or divide by a negative number, the inequality sign flips.

Example. −3x + 6 ≤ 0 ⇒ −3x ≤ −6 ⇒ x ≥ 2 ⇒ x ∈ [2, +∞)

Quadratic inequalities

1. Make the coefficient of x² positive (if needed, multiply by −1 and flip the sign).
2. Find the roots of the quadratic using the discriminant.
3. The parabola opens upwards: between the roots the quadratic is negative, outside them it is positive.

Example. x² − 3x − 4 < 0. Roots: x = −1 and x = 4. Answer: x ∈ (−1, 4).
For x² − 3x − 4 ≥ 0 the answer is x ∈ (−∞, −1] ∪ [4, +∞).

Absolute value inequalities

• |f(x)| < c (c > 0) ⇔ −c < f(x) < c — a double inequality;
• |f(x)| > c (c > 0) ⇔ f(x) < −c or f(x) > c;
• if c < 0, then |f(x)| < c has no solutions and |f(x)| > c holds for every x.

Example. |2x − 1| < 5 ⇔ −5 < 2x − 1 < 5 ⇔ −2 < x < 3 ⇒ x ∈ (−2, 3)

Writing the answer

A round bracket ( ) means the endpoint is not included (strict sign < or >), a square bracket [ ] means it is included (≤ or ≥). Infinity always gets a round bracket. The symbol ∪ joins intervals.