Explanation

In this question, as in every Data Sufficiency question, we start with the maxim, the mantra, "Separately First." we need to know the absolute value of w. That means the distance of w from zero - basically the positive "version" of a positive or negative number. Statement (1) gives two possibilities for w: and . Narrowing to two possibilities doesn't count as knowing the "value" on these questions--only one value. But the absolute value of 4 and of -4 are both 4. There is exactly one possible absolute value of w, given Statement (1), so statement (1) is Sufficient.

On to Statement (2), forgetting about Statement (1). We will paraphrase this statement. It says that the negative of the absolute value of w equals w. The absolute value of w is the distance from zero, a positive number, so the negative of that is a negative number, which is w. So it's basically saying that w is a negative number. But we don't know whether it's -3, -4, -10, what. We are looking at (2) alone, not in conjunction with (1). So Statement (2) alone is insufficient.

The correct answer is (A).