Inequality Solver

How to Solve a Linear Inequality

A linear inequality looks almost identical to a linear equation, written as ax + b compared to c using <, >, ≤, or ≥ instead of an equals sign. Where an equation like 2x + 3 = 9 has exactly one solution, an inequality like 2x + 3 < 9 has a whole range of solutions: every value of x that makes the statement true. Solving one means isolating x the same way you would in an equation, using the same basic moves of adding, subtracting, multiplying, and dividing on both sides.

There is exactly one rule that differs from solving equations: multiplying or dividing both sides by a negative number flips the direction of the inequality. Compare 2 < 5 with what happens when you multiply both sides by −1: −2 is not less than −5, it is greater than −5, so the sign must flip to −2 > −5 to stay true. The same logic applies whenever you divide by a negative coefficient. Forgetting this single step is the most common mistake when solving inequalities, so the calculator above flags it explicitly whenever it applies.

Once x is isolated, the solution can be written two ways: inequality notation, like x > 3, or interval notation, like (3, ∞), where a parenthesis marks a strict boundary that is not included and a bracket marks one that is. Two special cases can also occur: if the variable cancels out entirely, the inequality is either true for every real number or false for all of them, meaning no solution exists at all.

The number line above makes these boundaries visible. An open circle marks a strict boundary excluded from the solution, matching the parenthesis in interval notation, while a filled circle marks a boundary that is included, matching a square bracket. The shaded, arrow-tipped ray then shows every solution extending toward infinity in the correct direction, so you can see at a glance whether x is bounded above, bounded below, or unrestricted entirely.