If the fraction m/n is negative, which of the following CANNOT be true

APPROACH #1: Number sense and inequality manipulation
If \(\frac{m}{n}\) is negative, then we know that one value (m or n) is positive, and the other value is negative.
When I scan the answer choices, answer choice D looks like a good contender.

If \(\frac{m}{n}\) is negative, we know that m ≠ 0 and n ≠ 0.
Next, if n ≠ 0, then we know that \(n^2\) is positive.
If \(n^2\) is positive, we can safely take the inequality \(mn^3\) > 0 and divide both sides by \(n^2\) to get: \(mn>0\)
If \(mn>0\), then we know that EITHER m and n are both positive OR m and n are both negative.
This, however, contradicts our earlier conclusion that one value (m or n) is positive, and the other value is negative.
Answer: D

APPROACH #2: Using counter-examples to eliminate 4 of the answer choices
The question asks, "which of the following CANNOT be true?"
So if we can find values for m and n that make a statement true, we can eliminate that answer choice.

A. \(\frac{n}{m}\)>\(\frac{m}{n}\)
If n = 1 and m = -2, then it's true that \(\frac{n}{m}\)>\(\frac{m}{n}\)
ELIMINATE A.

B. mn < 0
If n = 1 and m = -2, then it's true that mn < 0
ELIMINATE B

C. n – m > 0
If n = 1 and m = -2, then it's true that n – m > 0
ELIMINATE C

E. m – n > 0
If n = -1 and m = 2, then it's true that m – n > 0
ELIMINATE E

By the process of elimination, the correct answer is D

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