How to Solve Inequalities

An inequality is a mathematical statement that compares two expressions using a symbol other than equals. Instead of finding one exact value, solving an inequality means finding every value that makes the comparison true — a range of numbers rather than a single answer.

This guide covers every major inequality type in the order students typically encounter them: one-step, two-step, multi-step, compound, absolute value, quadratic, and rational. Each section explains the method, shows a worked example, and notes the one rule that most commonly causes errors for that type.

The single most important rule across all inequality types is the sign flip: when you multiply or divide both sides of an inequality by a negative number, the inequality symbol reverses direction. This rule has no equivalent in equation solving, and it applies to every inequality type in this guide.

If you need a quick refresher on inequality symbols, want to rewrite answers in interval notation, or want a visual companion for the final answer, the guide on how to graph inequalities on a number line fits naturally alongside this pillar article.

A one-step inequality requires a single operation to isolate the variable. The operation is addition, subtraction, multiplication, or division.

For addition and subtraction, the symbol never changes. For multiplication and division, the symbol stays the same when the number is positive and flips when the number is negative.

A two-step inequality requires two operations: first remove the constant, then remove the coefficient. The order matters — removing the constant first keeps the numbers cleaner.

The sign flip applies only at the second step (division or multiplication), not at the first step (addition or subtraction).

Multi-step inequalities involve combining like terms, distributing, or moving variable terms before isolating x. The solving order follows the same logic as multi-step equations: distribute, combine like terms, collect variable terms on one side, then isolate.

The sign flip rule still applies only at the final division or multiplication step.

A compound inequality combines two conditions. AND inequalities require both conditions to be true simultaneously — the solution is the overlap (intersection) of the two sets. OR inequalities require at least one condition to be true — the solution is the union of both sets.

Chained inequalities like -1 < 2x + 3 ≤ 9 are AND inequalities written in compact form. Solve by applying the same operation to all three parts simultaneously.

Absolute value measures distance from zero. |A| < k means A is within distance k of zero, which rewrites as -k < A < k (an AND inequality). |A| > k means A is more than distance k from zero, which rewrites as A < -k or A > k (an OR inequality).

Always check the right-hand side first. If k is negative, |A| < k has no solution (absolute value cannot be negative). If k is zero, |A| > 0 is true for all x ≠ 0.

A quadratic inequality contains a squared variable term. The standard method is the sign chart: find the roots of the corresponding equation, then test the sign of the expression in each region between the roots.

The solution is either the region between the roots (for < or ≤) or the two outer regions (for > or ≥), depending on whether the parabola opens upward or downward.

A rational inequality has a variable in the denominator. The method is similar to quadratic inequalities — find the critical values (zeros of the numerator and undefined points of the denominator), then test the sign of the expression in each region.

Never multiply both sides by the denominator directly, because the denominator's sign is unknown and could flip the symbol. Use the sign chart method instead.

Every inequality type in this guide follows the same sign flip rule: multiplying or dividing both sides by a negative number reverses the inequality symbol. This is the only operation that changes the symbol. Addition, subtraction, and multiplication or division by a positive number never flip it.

The reason is geometric. When you multiply both sides by -1, you reflect every value across zero on the number line. A value that was to the right of the boundary is now to the left. The relationship between the two sides reverses, so the symbol must reverse to remain true.

In practice, the sign flip check is a one-second habit: before writing the final answer after a division or multiplication step, ask whether the number you divided or multiplied by was negative. If yes, flip the symbol. If no, keep it.

Each inequality type in this guide has a dedicated calculator page. Use the calculator that matches the type of problem you are working on to see the step-by-step solution, number line graph, and interval notation for your specific input.

For quick linear practice, move between the one-step inequality calculator, two-step inequality calculator, and multi-step inequality calculator. For split answers and interval logic, compare the output with the compound inequality calculator.

When the structure changes, switch tools rather than forcing a linear method onto the wrong problem. Use the absolute value inequality calculator, quadratic inequality calculator, or rational inequality calculator when the expression itself demands a sign-chart method or a case split.