hazelnut wrote:
If mn ≠ 0, is m > n?
(1) 1/m < 1/n
(2) m^2 > n^2
Official Explanation:The constraint in the question stem indicates that neither m nor n equals zero.
(1) INSUFFICIENT: You can solve algebraically/theoretically or you can Test Cases. If you solve algebraically, be careful:
you have to account for multiplying the inequality by a negative.
If m and n are both positive, then \(m > n\).
If m and n are both negative, the sign flips twice, so \(m > n\) again.
If only one is negative, then the sign flips once and \(m < n\). In this case, m must be the negative number, since any positive
is greater than any negative.
Alternatively, Test Cases:
If m = 3 and n = 2, then statement (1) is true and the answer to the question is Yes, \(m > n\).
If m = −3 and n = 2, then statement (1) is true and the answer to the question is No, m is not greater than n.
(2) INSUFFICIENT: This statement indicates nothing about the signs of the two variables. Either one could be positive or
negative.
(1) AND (2) INSUFFICIENT. If you are solving algebraically, test the scenarios that you devised for statement (1).
If m and n are both positive, then \(m > n\) and \(m^2 > n^2\). Both statements allow this scenario.
If m and n are both negative, then \(m > n\) but and \(m^2\) is not greater than \(n^2\). Discard this scenario, since it makes statement
(2) false.
If m is negative and n is positive, then \(m < n\). It could also be true that \(m^2 > n^2\), as long as m's magnitude is larger than
n's. If you're not sure Test Cases (see below).
Alternatively, test cases. Start by testing whether the cases you already tried for statement (1) also apply to statement (2).
If m = 3 and n = 2, then \(m > n\) and m² > n². Both statements allow this scenario.
If m = −3 and n = 2, then \(m < n\) and m² > n². Both statements allow this scenario.
Because there are scenarios in which \(m > n\) and \(m < n\), both statements together are still insufficient to answer the question. If
you forgot to account for the positive and negative cases, you may end up with (A) or (D) as your (incorrect) answer.
The correct answer is
(E).
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