Linear inequalities in one or two variables

What the SAT tests here

Inequalities describe a range of values instead of a single answer. You will solve them, translate phrases like at least and at most, find the greatest or least possible value in a story, and decide which points satisfy one or two inequalities.

Solving: the same moves as an equation, with one rule

Add, subtract, multiply and divide exactly as you would in an equation. When you multiply or divide both sides by a negative number, flip the inequality sign.

• 3x − 4 > 11 → 3x > 15 → x > 5.
• −2x + 6 ≤ 14 → −2x ≤ 8 → divide by −2 and flip → x ≥ −4.

Words to symbols

Phrase	Symbol
at least, no less than, minimum	≥
at most, no more than, maximum	≤
more than, exceeds	>
less than, fewer than	<

Compound inequalities

−3 < 2x + 1 ≤ 7. Do the same step to all three parts: subtract 1 to get −4 < 2x ≤ 6, then divide by 2 to get −2 < x ≤ 3. The integers in this range are −1, 0, 1, 2 and 3.

Worked example: the greatest whole number

A van can carry at most 1200 kg. The driver weighs 80 kg and each box weighs 30 kg. How many boxes can it carry?

• 30b + 80 ≤ 1200 → 30b ≤ 1120 → b ≤ 37.3.
• Boxes come in whole numbers, so the answer is 37. Round down for a maximum, even though 37.3 is closer to 37 than to 38 for a different reason: 38 boxes would break the limit.

Worked example: a minimum score

A student needs an average of at least 85 on four tests. The first three scores are 80, 84 and 88.

• (80 + 84 + 88 + s) / 4 ≥ 85 → 252 + s ≥ 340 → s ≥ 88.

Points and regions

A point satisfies an inequality if substituting it makes the statement true. For y > 2x − 1, the point (0, 1) works because 1 > −1, while (2, 2) fails because 2 > 3 is false.

For a system of inequalities, a point must satisfy every inequality. Is (1, 2) in the region y ≥ x and y ≤ 4 − x? Check both: 2 ≥ 1 is true and 2 ≤ 3 is true, so yes.

On a graph, a solid line means the boundary is included (≤ or ≥) and a dashed line means it is not (< or >).

Common traps

• Forgetting to flip after multiplying or dividing by a negative.
• Rounding the wrong way in a maximum or minimum story. A maximum rounds down, a minimum rounds up.
• Testing only one inequality of a system.
• Reading at least as greater than. At least includes the number itself.

Calculator tip

Type an inequality such as y > 2x − 1 into the built-in graphing calculator to see the shaded region. Add points to check them, or add a second inequality to see where the two regions overlap.

Quick summary

• Solve like an equation; flip the sign when multiplying or dividing by a negative.
• At least is ≥, at most is ≤.
• Maximum stories round down, minimum stories round up.
• A point solves a system only if it passes every inequality.