Linear inequalities in one or two variables

An inequality tells you not one answer but a whole range of answers. Instead of 𝑥 =5, you get something like 𝑥 >5, meaning every number greater than 5 works.

In one variable, you solve an inequality almost exactly like an equation: add, subtract, multiply, divide to isolate the variable. The single critical difference is the flip rule — whenever you multiply or divide both sides by a negative number, reverse the direction of the inequality sign.

In two variables, an inequality like 𝑦 <2𝑥 +1 doesn't describe a line but an entire region of the coordinate plane. The boundary line splits the plane into two half-planes, and one of those half-planes contains every point that satisfies the inequality. A dashed boundary means the line itself isn't included (strict inequality); a solid boundary means it is.

On the SAT, inequality questions often come wrapped in real-world constraints — budgets, minimums, capacity limits. The core skill is the same: translate the situation into an inequality, solve or graph it, and interpret what the solution set means in context.