Inequality Solver
Enter a linear inequality in x, such as 2x + 3 < 11 or 5 − 3x ≥ 2x + 1. The solver isolates x, reports the solution as an interval and on a number line, and shows every step — including the sign flip when you divide by a negative.
Use <, >, <= or >= (≤ and ≥ also work). Only the variable x is supported.
What is an inequality?
An inequality is a statement that two expressions are not necessarily equal — one is smaller than, larger than, or at most/at least the other. Instead of an equals sign it uses one of <, ≤, >, ≥. A solution is any value that makes the statement true; unlike an equation, that solution set is usually a whole range of numbers rather than a single value.
Solving inequalities, piece by piece
Each card explains one thing this tool does with your input — how it isolates x, when it must flip the inequality sign, and how it renders the solution set as an interval and a number line.
Isolate x — the same moves as an equation
Move constants to one side and gather x on the other. Adding or subtracting the same value from both sides, and multiplying or dividing by a positive number, all keep the inequality sign as it is.
Multiply or divide by a negative — sign flips
Multiplying both sides by a negative number reverses the order of the two values on the number line, so the inequality symbol has to flip to keep the statement true.
Interval & number-line notation
The tool writes the solution as an interval and draws it on a number line. A round bracket ( or open circle marks a strict boundary; a square bracket [ or filled circle marks ≤ / ≥.
Features of this calculator
- Accepts inequalities with <, >, ≤, ≥ (or <= and >=) in either direction
- Handles x on either side and combines like terms automatically
- Reports the answer as x < a (or ≤, >, ≥), in interval notation, and on a number line
- Uses open circles for strict inequalities and closed circles for non-strict
- Explicitly flags the step where the inequality sign flips
- Detects always-true and never-true statements after the x-terms cancel
Frequently asked questions
+Does the sign flip when I add or subtract?
No. Adding or subtracting the same value from both sides never changes the inequality's direction. Only multiplication or division by a negative number does.
+How do I solve a compound inequality like 1 < 2x + 3 ≤ 7?
Split it into two parts joined by AND — 1 < 2x + 3 and 2x + 3 ≤ 7 — solve each separately with this calculator, and take the intersection of the two solution intervals. In this example you get 2x > −2 and 2x ≤ 4, giving −1 < x ≤ 2, or (−1, 2].
+What if x cancels out completely?
Then the inequality reduces to a plain numeric statement. If it's true (like 3 < 7) every real number is a solution; if it's false (like 4 < 1) there are no solutions. The solver reports both cases clearly.
+How do I write the solution in interval notation?
Round brackets ( and ) mean the endpoint is excluded (strict < or >). Square brackets [ and ] mean it's included (≤ or ≥). Infinity always uses a round bracket because it isn't a number you can reach.