Given this is a Data Sufficiency question, and that it is in the form of "Yes/No", we know that we can eliminate a statement if there are cases that provide a "yes" and "no" given the information.
With this in mind, make sure to write down the information provided and then reflect on what the statements imply. We know that x and y could either be negative or positive, so there are opportunities for the same sign for both numbers, and opposite signs for both numbers. If X and Y are both negative, then the answer will be yes. If X and Y are both positive, then the answer will be yes. If one is positive and the other is not positive, it depends. This depends part is where we are going to have to see if we get any additional information that can help solve.
Statement 1 - the absolute value of x being greater than the absolute value of y, has a few potential cases.
1) X and Y both positive -- X larger than Y (answer to question is yes).
2) X and Y are both negative, and X is more negative than Y (Yes to question).
3) X is negative and Y is positive. Absolute value of X still has to be greater than Y -- so, for example, -3 for X and 2 for Y, we have 1 for the equation on the left and and then -5 for equation on right --> gives us NO. Shows that if X is negative and Y is positive, and the absolute value of X is greater than Y, the answer will be no.
We can cross out A and D from this.
Statement 2: |x-y| < |x| --> this implies that X and Y must both be positive, and that y must be less than X.
With this in mind, we can answer the question.
This problem requires knowledge of absolute value relationships, which
Bunuel has a great collection for preparation purposes. Memorizing these relationships will allow you to solve problem substantially faster than having to reason and test cases for each problem.