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Re: If the fraction m/n is negative, which of the following CANNOT be true
[#permalink]
03 Sep 2021, 06:00
Expert Reply
Top Contributor
Bunuel wrote:
If the fraction \(\frac{m}{n}\) is negative, which of the following CANNOT be true?
A. \(\frac{n}{m}\)>\(\frac{m}{n}\)
B. mn < 0
C. n – m > 0
D. \(mn^3\) > 0
E. m – n > 0
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
Re: If the fraction m/n is negative, which of the following CANNOT be true
[#permalink]
03 Sep 2021, 06:14
Expert Reply
If m/n < 0, then multiplying by n^4 on both sides, we find mn^3 < 0, so answer D must be false.
Note that it's safe to multiply by n^4 on both sides, because n^4 must be positive here (it's an even power, and it can't be zero), so we don't need to worry about whether to reverse the inequality. _________________
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