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.

Any inequality as text

In the fourth mode you type the inequality as text. Polynomials and fractions are solved exactly. Exponential, logarithmic, trigonometric and radical inequalities are solved numerically, and the solution says so. For trigonometric ones the calculator finds the period and writes the answer with k ∈ ℤ.

Example. sin x ≥ 1/2 gives x ∈ [π/6 + 2πk, 5π/6 + 2πk], k ∈ ℤ.