Solving Inequality Calculator

Enter the numbers of your inequality and get the solution with every step shown — including when to flip the inequality sign — the answer in inequality and interval notation, and a number line graph with open or closed circles. Works for linear, compound, absolute value and quadratic inequalities.

How to solve a linear inequality

Solve an inequality almost exactly like an equation: add or subtract the same amount on both sides, and multiply or divide both sides by the same number. The one extra rule is that multiplying or dividing both sides by a negative number reverses the inequality sign. For example, −3x + 4 < x − 8 becomes −4x < −12 after subtracting x and 4, and dividing by −4 flips the sign: x > 3. The calculator states each step and highlights when the sign flips.

Why the sign flips

Multiplying by a negative number reverses the order of numbers on the number line. We know 2 < 5, but −2 > −5. So if −4x < −12, dividing by −4 must turn “<” into “>” to stay true: x > 3. You can always check a solution by substituting a value: x = 4 gives −3(4) + 4 = −8, and 4 − 8 = −4; −8 < −4 is true, so 4 is a solution.

Types of inequalities the calculator solves

Interval notation and number lines

Round brackets and open circles mean the endpoint is not included (< or >); square brackets and filled circles mean it is included (≤ or ≥). Infinity always takes a round bracket.

Solving absolute value inequalities

• |u| < c (with c > 0) means u is within c of zero: −c < u < c. The solution is one interval.
• |u| > c means u is more than c away from zero: u < −c or u > c. The solution is two separate rays.
• If c is negative, |u| < c has no solution and |u| > c is true for every x, because an absolute value is never negative.
• A memory aid: “less thAND” (one connected piece) and “greatOR” (two pieces).

Solving quadratic inequalities

1. Rearrange so one side is zero: ax² + bx + c > 0.
2. Find the roots of ax² + bx + c = 0 by factoring or the quadratic formula.
3. The roots split the number line into regions. The parabola’s sign is constant in each region.
4. If a > 0, the parabola opens upward: it is negative between the roots and positive outside. If a < 0, the reverse.
5. Choose the regions that match the inequality sign and write the answer.

If there are no real roots (the discriminant is negative), the quadratic is always positive (a > 0) or always negative (a < 0), so the answer is either all real numbers or no solution.

Worked example: quadratic

Solve x² − x − 6 > 0. The roots of x² − x − 6 = 0 are x = −2 and x = 3 (it factors as (x − 3)(x + 2)). Because a = 1 > 0, the parabola opens upward and is positive outside the roots. The solution is x < −2 or x > 3, or (−∞, −2) ∪ (3, ∞) in interval notation. Enter a = 1, b = −1, c = −6 and “>” in the quadratic mode to check.

Common mistakes

• Forgetting to flip the sign when dividing by a negative.
• Flipping the sign when subtracting a negative number (you only flip when multiplying or dividing).
• Writing “−2 > x > 3” for a two-part answer; use “or”.
• Using a filled circle for strict inequalities.
• Dividing by a variable whose sign you do not know.

Checking your answer

1. Pick a number inside your solution set and substitute it into the original inequality — it should make the statement true.
2. Pick a number outside the solution set — it should make the statement false.
3. Check the boundary value itself: it should satisfy the inequality only if the sign includes “or equal to”.

For −3x + 4 < x − 8 with solution x > 3: x = 4 gives −8 < −4 (true), x = 0 gives 4 < −8 (false), and x = 3 gives −5 < −5 (false), so 3 is correctly excluded.

Inequalities in real life

Inequalities describe limits and ranges: a budget that must not exceed a certain amount, a speed limit, the number of tickets needed to break even, a safe temperature range for food, or a weight limit for a lift. Writing the situation as an inequality — 12x + 150 ≤ 600 for the number of $12 tickets that keeps costs within a $600 budget including a $150 fee — and solving it gives a clear answer: x ≤ 37.5, so at most 37 tickets.

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