PSAT Math Multiple-Choice Question 37: Answer and Explanation

Question: 37

2 < 24

What is the sum of all the integer solutions to the inequality above?

• A. 7
• B. 9
• C. 10
• D. 14

Correct Answer: B

Explanation:

B

The question asks for the sum of the integer solutions to an inequality that includes an absolute value. Inequalities with an absolute value can be tough to set up: there are two parts, and it's not always easy to remember how to deal with the negative values. Instead, try plugging in some integers. Make y = 0. The inequality becomes 2 |0(4) - 5| < 24, which becomes 2 |0 - 5| < 24. Continue following the order of operations to get 2 |-5| < 24, or 2(5) < 24. It is true that 10 < 24, so 0 is one of the solutions. Now plug in y = 1, which results in 2 |4–5| < 24, or 2(1) < 24. That's true, too, so 1 is one of the solutions. Keep going: plugging in y = 2 results in 2(3) < 24, y = 3 results in 2(7) < 24, and y = 4 results in 2(9) < 24. All of these are true, but plugging in y = 5 results in 2(15) < 24, which isn't true. So 4 is the greatest integer solution. Now, try plugging in negative integers. If y = -1, the inequality is 2(9) < 24, which is true, but if y = -2, the inequality is 2(13) < 24, which is not true. Therefore, -1 is the smallest integer solution. The integers that satisfy the inequality are -1, 0, 1, 2, 3, and 4, and their sum is 9. The correct answer is (B).