Linear Inequalities

Unit 1 · Algebra

What the SAT asks here

Linear inequalities work like linear equations with one critical extra rule: the inequality sign flips when you multiply or divide both sides by a negative number. The SAT also asks you to set up budget-style systems from short word problems.

The flip rule

Multiplying or dividing by a negative flips the sign

-2x > 6 \;\Longrightarrow\; x < -3

System of inequalities — geometric meaning

(x, y) must satisfy every inequality at once

Solution region = overlap of all shaded half-planes

SAT Tip: After you isolate the variable, ask yourself once: 'Did I divide by a negative?' If yes, double-check the direction of the sign before you commit to an answer.

Caution: Non-strict inequalities (≤, ≥) include the boundary; strict ones (<, >) do not. The SAT loves to ask whether a boundary point counts.

Type 1

Solve a single inequality

Isolate the variable using the same steps as a linear equation, then flip the sign if you divided by a negative.

Example 1

Solve −3x + 5 ≤ 11 for x.

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Type 2

Test a point in a system

Substitute the point into every inequality. A point only satisfies the system when every inequality is true.

Example 2

Is (2, 1) a solution to the system y < 3x − 1 and y ≥ x − 2?

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Type 3

Word problem setup

Translate budget, capacity, or quantity constraints into inequalities. Don't forget non-negativity conditions when the variables represent counts or money.

Example 3

A student has at most $50 to spend on notebooks ($4 each) and pens ($2 each). They want at least 15 items total. Write a system describing valid purchases of n notebooks and p pens.

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