Is x > y ? (1) 6x ＞ 5y (2) xy ＜ 0

Official Solution:


We must determine if \(x\) is greater than \(y\).

Statement 1 tells us that \(6x \gt 5y\). To isolate \(x\), we divide both sides by 6, leaving \(x \gt \frac{5}{6} y\). From this inequality, there is no way to tell if \(x \gt y\) or if \(\frac{5}{6}y \lt x \lt y\). For example, if \(x = 13\) and \(y = 12\), then statement 1 becomes \(13 \gt \frac{5}{6}(12)\), or \(13 \gt 10\), which is true, and \(x \gt y\). If, however, \(x = 11\) and \(y = 12\), then statement 1 becomes \(11 \gt \frac{5}{6}(12)\), or \(11 \gt 10\), which is also true, only now \(y \gt x\). In other words, \(x\) can be greater than or less than \(y\). Statement 1 is NOT sufficient to answer the question. Eliminate answer choices A and D. The correct answer choice is B, C, or E.

Statement 2 tells us that the product of \(x\) and \(y\) is less than 0. This can only be the case if one, but not both, of \(x\) and \(y\) is negative, and one, but not both, of \(x\) and \(y\) is positive. Yet without further information, there is no way to tell which one is negative and which one is positive, and thus no way to tell if \(x \gt y\). Statement 2 is NOT sufficient. Eliminate answer choice B. The correct answer must be either C or E.

Considering the statements together, we know that one of \(x\) and \(y\) is positive and one is negative, and that \(x \gt \frac{5}{6}y\). Since no negative number can be greater than the product of a positive fraction and a positive number, we know that \(x\) must be positive and that \(y\) must be negative. Thus, \(x \gt y\). The statements together are sufficient to answer the question.


Answer: C.