Explanation

This question is rather simple-looking. Let's head to the data statements, evaluating them separately at first.

Statement (1) tells us that . That means . Does that fact allow us to answer definitively? It may appear to allow us to answer the question definitively in the negative. To be sure, we can analyze by cases.Case I: . That is permitted because . In this case, it is true that , because , so the answer would be "yes." Case II: . That case is permitted, as the sum is still greater than zero. In this case, is ? No; is false. We have obtained different answers from allowed cases, so we don't have sufficient information to answer the question definitively. Statement (1) is insufficient.

Statement (2) tells us that w to the v power is negative. The only way that could happen is if w itself is negative. In fact, this data statement allows our case , where it's true that , and the answer is "yes." Could v be less than w, which has to be negative? Yes; could be , a.k.a. . That's negative, so it's allowed by Statement (2), and it gives a "no" answer to , because -3 is more negative than -2 and is hence less than -2. So, again we have a "yes" and a "no" in permitted cases. Statement (2) is insufficient.

Combining the statements, we're tiring a little of constructing cases, so we prefer to use preexisting cases that are allowed by both statements. As we've seen, the case is allowed by both statements, and it gives us a "yes" to the question of . The other cases are no longer allowed. We look at Statement (1). Since w has to be negative now, v will have to be positive; that's the only way to get the sum above zero, as Statement (1) requires. And if w has to be negative and v has to be positive, that means that, in all allowed cases, v will be greater than w. We can answer the question definitively in the affirmative. The statements are sufficient together and the correct answer is (C).

This was a typical finish to a question in which rules, such as number properties rules, were in play in the question, but we didn't necessarily know what those rules were. What will happen is that you can analyze by cases and you may hit a moment when you notice a regularity. Here it was the fact that, when the statements were combined, v had to be positive and w negative. Keeping an eye out for conclusions that you can draw will be superior in general to attempting to analyze by cases through brute force.

Again, in this question, the correct answer is (C).