Inequality Solver

Q: When does the inequality sign flip?

Whenever both sides are multiplied or divided by a negative number, for example in −2x −3. Adding or subtracting never changes the sign.

Q: How do you write the solution as an interval?

From left to right with brackets: (−∞, 2) means all numbers below 2, [3, ∞) all numbers from 3 upward including 3. Several ranges are joined with the union sign ∪.

This inequality solver handles linear, quadratic and absolute value inequalities: pick the type, enter the numbers and read the solution as an inequality and an interval. Example: 2x + 3 ≤ 5x − 6 gives x ≥ 3 because the sign flips when dividing by −3.

Example: Type of inequality: Linear: a·x + b ⋚ c·x + d · Comparison: less than or equal (≤) · a (coefficient of x, left) 2 · b (number, left) 3 · c (coefficient of x, right) 5 · d (number, right) -6 → Result: Solution: x ≥ 3. Source: Wikipedia – Ungleichung (Äquivalenzumformungen). Updated: 2 October 2026.

Type of inequality

Comparison

a (coefficient of x, left)

You are solving for x in a·x + b ⋚ c·x + d.

b (number, left)

c (coefficient of x, right)

d (number, right)

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Result

Result: Solution: x ≥ 3
Solution set: Item Value
Inequality 2x + 3 ≤ 5x − 6
Solution as inequality x ≥ 3
Solution as interval [3, ∞)
Steps: Move all x terms to the left and all numbers to the right: -3·x ≤ -9.
Dividing by -3, a negative number: the inequality sign flips to ≥.
Solution: x ≥ 3
Check: For x = 3, “left side minus right side” equals 0; 0 ≤ 0 is true.
For x = -100, “left side minus right side” equals 309; 309 ≤ 0 is false.; When you divide or multiply by a negative number, the inequality sign flips (≤ becomes ≥). This is the most common source of mistakes.

Item	Value
Inequality	2x + 3 ≤ 5x − 6
Solution as inequality	x ≥ 3
Solution as interval	[3, ∞)

How it is calculated

An inequality is rearranged like an equation, with one exception: multiplying or dividing by a negative number flips the comparison sign. The solution is usually not a single value but a range of numbers, written as an inequality (x ≥ 3) or as an interval ([3, ∞)).

The three types

For a linear inequality a·x + b ⋚ c·x + d, move all x terms to the left and all numbers to the right, then divide by (a − c). For a quadratic inequality a·x² + b·x + c ⋚ 0, find the roots with the discriminant and read from the parabola where it lies above or below the x-axis. For an absolute value inequality |a·x + b| ⋚ c, split the absolute value into two cases: the term lies between −c and c, or outside that range.

Example with a sign flip

2x + 3 ≤ 5x − 6 becomes −3x ≤ −9. Dividing by −3 flips the sign: x ≥ 3. Checking x = 4 confirms it (11 ≤ 14), while x = 0 fails (3 ≤ −6 is false). The solver shows this check below the result.

Round brackets in an interval exclude the endpoint, square brackets include it.
With strict signs (< and >) the endpoint is never part of the solution.
The result can also be “all real numbers” or “no solution”, for example for x² + 1 < 0.

Frequently asked questions

When does the inequality sign flip?

Whenever both sides are multiplied or divided by a negative number, for example in −2x < 6, which gives x > −3. Adding or subtracting never changes the sign.

How do you solve quadratic inequalities?

First find the roots of the related equation. For an upward-opening parabola (a > 0) the expression is negative between the roots and positive outside; for a < 0 it is the other way round.

How do you write the solution as an interval?