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Linear inequalities in one or two variables — SAT practice questions

Linear inequalities in one or two variables on the SAT Math test cover solving inequalities, rewriting them into equivalent forms, and deciding which points satisfy a two-variable constraint. A student must isolate the variable carefully, especially when multiplying or dividing by a negative, translate an at-most spending limit into an inequality, and test ordered pairs against the boundary. Items often ask which point is not a solution, which inequality models a word problem, or which form is equivalent. Choices look like near-matching inequalities or coordinate pairs that flip the sign or sit on the wrong side of the line.

  • Section: Math 44 questions on the paper
  • 111 practice questions in the app

Sample questions

Question 1

To reserve spots on a bicycle tour, a guide must pay a $43.50 fee plus $14.80 per rider. The guide can spend at most $283. If rr is the number of riders, which inequality represents this situation?

  1. A43.50+14.80r≥28343.50+14.80r\ge283
  2. B14.80+43.50r≤28314.80+43.50r\le283
  3. C43.50+14.80r≤28343.50+14.80r\le283
  4. D14.80+43.50r≥28314.80+43.50r\ge283
Show answer

Answer: C — 43.50+14.80r≤28343.50+14.80r\le283

The reservation cost is 43.50 plus 14.80 for each of rr riders. Because the guide can spend at most 283, that cost must satisfy 43.50+14.80r≤28343.50+14.80r\le283. The first choice reverses the inequality. The second and fourth choices swap the fixed fee with the per-rider charge.

Question 2

2x−y≤52x - y \le 5

Which of the following points is NOT a solution to the given inequality?

  1. A(0,0)(0, 0)
  2. B(1,−2)(1, -2)
  3. C(6,−2)(6, -2)
  4. D(−3,1)(-3, 1)
Show answer

Answer: C — (6,−2)(6, -2)

Testing (6,−2)(6,-2): 2(6)−(−2)=142(6)-(-2)=14, which is not ≤5\le5, so this point fails; the other three points each satisfy the inequality.

Question 3

1<3x−51<3x-5

Which inequality is equivalent to the given inequality?

  1. Ax>2x>2
  2. Bx≥2x\ge 2
  3. Cx<2x<2
  4. Dx>43x>\dfrac{4}{3}
Show answer

Answer: A — x>2x>2

Adding 5 to both sides gives 6<3x6<3x. Dividing both sides by 3 gives 2<x2<x, which is equivalent to x>2x>2.

Practice 111 Linear inequalities in one or two variables questions in the app

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