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Algebra

SAT linear inequalities: translate “at least” and “at most” correctly

Most inequality mistakes happen in translation or when dividing by a negative. Learn the phrase-to-symbol table, see solutions on a number line, and solve the word problems the SAT likes to ask.

SPSAT Practice Editorial7 min read

The short version

What you'll take away

  • “At least” means ≥ and “at most” or “no more than” means ≤.
  • Solve inequalities like equations, but flip the inequality sign when you multiply or divide by a negative number.
  • In word problems, round to the whole number that still satisfies the constraint, not always to the nearest whole number.
In this article

Translate the words first

Inequality word problems hinge on a few phrases. Getting the direction and whether the endpoint is included right matters more than any algebra step.

See the skill

Phrase to symbol

at least, a minimum of, no less than

≥

“At least 20 students” → s ≥ 20

at most, a maximum of, no more than

≤

“At most $35” → c ≤ 35

more than, exceeds

>

“More than 8 hours” → h > 8

fewer than, less than

<

“Fewer than 5 errors” → e < 5

Use the visual as a reasoning aid: tie each answer choice to the rule, relationship, or evidence it depends on.

Solve like an equation, with one twist

Add, subtract, multiply, and divide on both sides, just as you would for an equation. The twist: multiplying or dividing by a negative number reverses the inequality. For \(\displaystyle -3x + 5 > 11\): subtract 5 to get \(\displaystyle -3x > 6\), then divide by −3 and flip the sign: \(\displaystyle x < -2\).
Check with a number. Try \(\displaystyle x = -3\): \(\displaystyle -3(-3) + 5 = 14\), and \(\displaystyle 14 > 11\) is true. Try \(\displaystyle x = 0\): 5 is not greater than 11, so 0 is correctly excluded.

See the skill

Solutions of −3x + 5 > 11

x < −2

closed: included (≤, ≥) open: not included (<, >)
The open circle shows that −2 itself is not a solution, because −3(−2) + 5 = 11 is not greater than 11.

Compound inequalities

A compound inequality traps a quantity between two values. Do the same operation to all three parts. For \(\displaystyle -1 < 2x + 3 \le 9\): subtract 3 to get \(\displaystyle -4 < 2x \le 6\), then divide by 2 to get \(\displaystyle -2 < x \le 3\).

See the skill

Solutions of −1 < 2x + 3 ≤ 9

−2 < x ≤ 3

closed: included (≤, ≥) open: not included (<, >)
The left end is open (<) and the right end is closed (≤), matching the original symbols.

Word problems: the right whole number

A phone plan costs $20 per month plus $0.10 per text, and Jordan wants to spend at most $35. Then \(\displaystyle 20 + 0.10t \le 35\), so \(\displaystyle 0.10t \le 15\) and \(\displaystyle t \le 150\). Jordan can send up to 150 texts.
When the answer must be a whole number, round in the direction that keeps the inequality true. If the algebra gives \(\displaystyle b \le 22.7\), the greatest possible whole number is 22, not 23.

Your turn

Quick practice

A delivery van can carry at most 1,200 pounds. The driver weighs 180 pounds, and each box weighs 45 pounds. What is the greatest number of boxes the van can carry with the driver?

Hint: Include the driver’s weight, then round down.

Choose one answer

Two-variable inequalities

Inequalities in two variables describe regions. If notebooks cost $3 and pens cost $5, and you can spend at most $60, then \(\displaystyle 3n + 5p \le 60\). To test whether a combination works, substitute it. On a graph, the boundary line is solid for ≤ or ≥ and dashed for < or >, and the shaded side holds the solutions.

Practice: try these before checking the answers

Watch for negative coefficients.
  • 1. Solve \(\displaystyle 4x - 7 \ge 13\).
  • 2. Solve \(\displaystyle 5 - 2x < 1\).
  • 3. Translate: “A team needs at least 11 players.”
  • 4. Is (2, 3) a solution of \(\displaystyle 3x + 2y \le 12\)?

Sources & further reading

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