i'm not completely sure, but think D is correct. Testing positive and negative cases for statement 1 shows it is insufficient. If x =3 & y=2 the statement will be true, but if x=-2, y=-3 the statement will be false.

Statement 2, x and y must have different signs. if you test different cases you also get different results, so insufficient.

If you combine both statements. Y must be negative and x positive, so testing numbers always makes the statement true.

Any tips for a fast way to do this problem?

The answer can't be D it would be C.

Consider when y = 0 and x = 1 that makes y < x true.

However | 1 - 0 | > | 1 | - | 0 | which equals 1 > 1 and that is false therefore statement 1 is insufficent.

Statement two says x or y cannont be zero and when you times them it's negative. Therefore x or y has to be negative. x = 1 and y = -2 that's 3 > 2. However when x = -1 and y = 1 you get 2 > 2 and therefore insuffient.

However if you combine both statements you can figure out you know x is postive has x > y and one of the numbers xy < 0 has to be negative so it makes y negative. So if x = 1 and y = -1 then 2 > 1. Suffient.

I hope that helps. Let me know if i need to clear anything up[/spoiler]

Apologies, D was a typo. Hopefully, my reply shows that I thought both statements alone were Insufficient, but combining them (C) would yield a sufficient answer.
Thanks!!