Inequalities

An inequality says one quantity is less than, greater than, or at most or at least as large as another. Where the equation 2x + 3 = 11 has the single solution x = 4, the inequality 2x + 3 < 11 is satisfied by every value less than 4. Solutions are written in interval form, described in words, or graphed on a number line, where an open circle marks a strict inequality (< or >) and a closed circle marks an inclusive one (≤ or ≥).

Inequalities are solved with the same moves as equations — add, subtract, multiply, divide on both sides — with one critical exception: multiplying or dividing both sides by a negative number reverses the direction of the inequality. Starting from −3x > 12 and dividing by −3 gives x < −4, not x > −4. Forgetting to flip the sign is the single most common error on test questions.

Several variations appear regularly. A compound inequality such as −2 < x + 1 ≤ 5 is solved by performing the same operation on all three parts, giving −3 < x ≤ 4. Absolute value inequalities split into cases: |x| < a becomes −a < x < a, while |x| > a becomes x < −a or x > a. A linear inequality in two variables graphs as a shaded half-plane bounded by a line, dashed for strict inequalities and solid for inclusive ones, and a system of such inequalities is solved by finding the region where the shadings overlap.

Inequalities are standard content across standardized math sections. The CLT tests them in algebra alongside systems of equations, the SAT tests linear inequalities and inequality modeling in advanced math, and the GRE covers inequalities together with the number line and absolute value, plus graphing inequalities in its quantitative reasoning section.