Algebra · Math
Linear inequalities
Linear inequalities and systems of them: solving, modelling a constraint in words, and identifying which points satisfy a region.
36 questions in the bank · 6 easier · 24 medium · 6 harder · this domain is ≈35% of the section
What to watch for
Multiplying or dividing by a negative reverses the inequality. In word problems, 'at most' means ≤ and 'at least' means ≥ — translate before you solve.
How the question is worded
These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.
- Which of the following describes all solutions to the inequality 8x + 5 ≥ 37?
- Which of the following describes all solutions to the inequality 8x + 10 ≥ 42?
- Which of the following describes all solutions to the inequality 8x + 12 ≥ 68?
- Which of the following describes all solutions to the inequality 6x + 4 ≥ 40?
Worked examples
easyWhich of the following describes all solutions to the inequality 8x + 5 ≥ 37?
Which of the following describes all solutions to the inequality 8x + 5 ≥ 37?
- A. x ≥ 37/8
- B. x ≥ 4
- C. x ≤ 4
- D. x > 4
Answer: B. x ≥ 4
Explanation
Subtract 5 from both sides: 8x ≥ 32. Divide by 8 — a positive number, so the direction stays the same: x ≥ 4. The ≥ is preserved throughout because every step used ≥.
mediumWhich of the following describes all solutions to the inequality −6x − 9 > 3?
Which of the following describes all solutions to the inequality −6x − 9 > 3?
- A. x < 2
- B. x > 2
- C. x > −2
- D. x < −2
Answer: D. x < −2
Explanation
Subtract −9: −6x > 12. Dividing both sides by −6 — a negative number — reverses the inequality: x < −2. Forgetting the flip is the single most common error on these.
hardWhat is the greatest integer value of x for which 4x − 11 < 46?
What is the greatest integer value of x for which 4x − 11 < 46?
Answer: 14
Explanation
Solve: 4x < 57, so x < 57/4. The greatest integer strictly below 57/4 is 14. Because the boundary is not an integer, no rounding subtlety arises — but always check whether the endpoint itself is allowed.