Linear inequalities on the SAT: lesson and practice
Solve an inequality exactly like an equation, with one extra rule: when you multiply or divide both sides by a negative number, flip the sign. In word problems, "at most" means ≤ and "at least" means ≥. For whole-number answers, round in the direction that keeps the inequality true.
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The one extra rule
−3x > 15
x < −5 (sign flipped when dividing by −3)
Translating words
| Words | Symbol |
|---|---|
| at most, no more than, maximum | ≤ |
| at least, no less than, minimum | ≥ |
| more than, exceeds | > |
| fewer than, below | < |
Example
A plan costs Rs 300 plus Rs 90 per GB. Hira can spend at most Rs 2,000. What is the most whole GB she can use?
90g ≤ 1,700
g ≤ 18.9, so 18 GB
Rounding up to 19 would break her budget.
Common mistakes
Sample questions with answers
1. Which describes all solutions to −2x + 4 > 10?
- x > −3
- x < −3 (correct)
- x < 3
- x ≤ −3
−2x > 6, and dividing by −2 flips the sign: x < −3.
2. What is the greatest integer x with x/3 − 4 < 2?
- 18
- 17 (correct)
- 6
- 5
x/3 < 6, so x < 18. The greatest integer is 17.
3. A lift carries at most 800 kg. A 70 kg operator carries 20 kg boxes. Which inequality fits the number of boxes b?
- 70 + 20b ≥ 800
- 20 + 70b ≤ 800
- 70 + 20b ≤ 800 (correct)
- 90b ≤ 800
"At most" means ≤, and each box adds 20 kg to the operator’s 70 kg.
Frequently asked questions
When do you flip the inequality sign?
Only when you multiply or divide both sides by a negative number.
What does "at most" mean in an SAT inequality?
Less than or equal to (≤). "At least" means greater than or equal to (≥).
How do you solve a compound inequality?
Do the same operation to all three parts at once, then read the range. Check whether each end is included.
How do I find a solution to a system of inequalities?
Test each answer choice in every inequality. A solution must satisfy all of them.