We are being asked to determine whether xy > 0. In other words, we want to know if xy is positive. There are two ways for xy to be positive - if x and y are both positive, or if x and y are both negative. If one of the variables is negative and the other is positive, xy will be negative, and if one of the variables is zero, xy will be zero. We know that x is not zero.
So to figure out whether each statement is sufficient, we should try to use logic or plug in smart numbers to get two answers: one where xy is positive (x and y have the same sign), and one where xy is negative (x and y have opposite signs) or zero (y = 0). If we can, the statement is insufficient. If we can't, the statement is sufficient.
Statement 1
This tells us that x is positive, but it doesn't tell us anything about y. So if y is positive, xy will also be positive. But if y is 0, xy will be zero, and if y is negative, xy will be negative. Insufficient.
Statement 2
Let's think about options here. If this statement is true, x and y can both be positive (ex: x = 2, y = 3 --> 1/2 < 3). This would make xy positive. However, we could also have a negative x and a positive y (ex: x = -2, y = 3 --> -1/2 < 3). This would make xy negative. We could also have a negative x and a zero y (ex: x = -2, y = 0 --> -1/2 < 0). This would make xy zero. Insufficient.
Both
If x is positive, then 1/x will also be positive. This means that if y is greater than 1/x, y must also be positive (not negative or zero). This would make xy positive. Sufficient.
