Problem Solving - Beatthegmat practice question

kuiper wrote:If 3x < 2y < 0, which of the following must be the greatest?

a. 2y - 3x
b. 3x - 2y
c. -(3x - 2y)
d. -(3x + 2y)
e. 0

OA given: D

We notice that -(3x - 2y) = -3x + 2y = 2y - 3x; i.e. the expression in answer choice A is equivalent to the expression in answer choice C. Therefore, neither A nor C can be the correct answer, and we eliminate both.

Next, notice that 3x - 2y is negative (since 3x < 2y); therefore, B cannot be the correct answer, either (because it is less than 0, which is answer choice E).

We need to decide between D and E. Notice that both 3x and 2y are strictly less than zero, and so is their sum. Since 3x + 2y is strictly less than zero, -(3x + 2y) is strictly greater than zero; which means D is the greatest.

Alternate Solution:

We can let x = -3 and y = -2 and check each answer choice.

Choice A: 2y - 3x = -4 - (-9) = 5

Choice B: 3x - 2y = -9 - (-4) = -5. Eliminate B.

Choice C: -(3x - 2y) = -(-5) = 5 (This is Choice B, with the sign reversed.) Because this is equal to Choice A, we can eliminate both A and C.

Choice D: -(3x + 2y) = -(-9 + (-4)) = -(-13) = 13. We compare this to Choice E and see that 13 is greater than 0. Choice D is the correct answer.

Answer: D