Inequality Solver

What does the inequality solver do?

It reads an inequality in one unknown the way you type it and finds the solution set. For linear inequalities it isolates the unknown and shows why dividing by a negative number reverses the sign. For quadratic, polynomial and rational inequalities it moves every term to the left, factors the expression, finds the critical points and builds a sign chart. The answer is given in interval notation, as an inequality and on a number line.

Method: the sign chart

1. Move every term to the left to get the form f(x) < 0. Fractions are combined into a single fraction.
2. Factor the numerator and the denominator.
3. Find the values that make the numerator zero (the expression is zero) and the values that make the denominator zero (the expression is undefined). These are the critical points.
4. The critical points split the number line into intervals. The sign cannot change inside an interval, so testing one value in each is enough.
5. Keep the intervals with the wanted sign. With ≤ or ≥ the zeros are included. Points where the denominator is zero are never included.

Example

Type x^2 - 5x + 6 < 0. The left side is (x − 2)(x − 3), so the critical points are 2 and 3. At x = 0 the value is 6 (positive), at x = 2.5 it is −0.25 (negative) and at x = 4 it is 2 (positive). The negative interval is wanted, so the solution is 2 < x < 3, or (2, 3) in interval notation.

Chains and systems

A double inequality such as -3 < 2x + 1 <= 7 can be typed on one line: the tool solves both parts and takes the intersection. In the system tab two inequalities go on separate lines. With "and" you get the values that satisfy both (intersection), with "or" the values that satisfy at least one (union).

Notation

A round bracket means the end point is excluded and a square bracket means it is included: (2, 3] stands for 2 < x ≤ 3. ∪ is a union, ∅ is the empty set and {2} is the single number 2. On the number line a filled dot is an included end point and a hollow dot an excluded one. The drawing is schematic: points are evenly spaced, not to scale.

Limits

Only polynomial and rational inequalities are solved. Square roots, absolute values, trigonometric, exponential and logarithmic inequalities are not supported. Coefficients must be whole numbers, fractions or decimals. Rational roots are exact and roots of quadratic factors are given with radicals. Irrational roots of higher-degree factors are written as decimal approximations, and the tool then says that the interval end points are approximate.