This looks like an inequality problem at first glance, but notice that the inequalities are all > 0. Whenever you see > 0 (positive) or < 0 (negative), think about the question from the point of view of positive and negative values or characteristics.
The question asks whether the sum of x and y is positive. If you know x and y are both positive, then the answer to this has to be Yes, but if x and y are both negative, then the answer has to be No. If one is positive and one is negative, then it depends on which value has the greater magnitude (distance from 0 on a number line).
(1) INSUFFICIENT: Test some cases to see what's possible. You're only allowed to try values for which statement (1) is true.
Case 1: If x = 3 and y = 2, then x – y > 0 and so is x + y. In this case, the answer is Yes, x + y > 0.
Case 2: If x = 2 and y = –3, then x – y > 0 but now x + y is less than 0, not greater than 0. In this case, the answer is No, x + y is not greater than 0.
Since it's possible to get both Yes and No answers, this statement is not sufficient to answer the question.
(2) INSUFFICIENT: First, rearrange the inequality given in the statement. x2 – y2 is one of the special quadratics and can also be written (x + y)(x – y). The statement establishes that (x + y)(x – y) > 0. In other words, these two quantities multiply to a positive value. Therefore, it must be the case that either (x + y) and (x – y) are both positive or that (x + y) and (x – y) are both negative. Since x + y could be either positive or negative, this statement is not sufficient to answer the question.
(1) AND (2) SUFFICIENT: The second statement establishes that (x + y) and (x – y) have to have the same sign—they're either both positive or both negative. The first statement says that (x – y) is positive. Therefore, (x + y) must also be positive, so the answer to the question is Always Yes, x + y > 0.
The correct answer is (C): Both statements together are sufficient to answer the question but neither statement works on its own.
