Linear Inequalities
Translating word problems into inequalities and finding valid ranges
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Same algebra as equations, but the answer is a range instead of a single number.
Why this matters
The SAT tests five distinct question types here: translating a constraint into an inequality, defining a valid range, checking whether a value satisfies the inequality, reading a graph, and solving for a maximum or minimum.
The five patterns
Translate
Turn a word problem about budgets, capacities, or goals into an inequality. Match each rate to the right variable and pick the correct direction: "at most" means ≤, "at least" means ≥.
›Range
Express a valid range using a compound inequality. Given a minimum and maximum, write a ≤ x ≤ b. Harder versions multiply a rate by both bounds or apply the triangle inequality.
›Verify
Test whether a point, row, or table of values satisfies an inequality. Substitute and check. For strict inequalities, the boundary value itself does not count.
›Interpret
Connect an inequality to its real-world meaning — what an ordered pair represents, what a term means, or which shaded graph matches the algebra.
›Solve
Set up and solve an inequality to find a maximum, minimum, or greatest integer. The key detail: round up for "at least" problems, round down for "at most" problems.
The biggest trap: rounding the wrong direction. If 130 students need buses that hold 28, you get 4.64… buses. You need 5, not 4. For "at least" constraints, always round up. For "at most" constraints, always round down. The SAT offers both options every time.