Is |x-y| > |x| - |y| ?
1) y < x
2) xy < 0

One approach is to plot the distances on a NUMBER LINE.

|x|= the distance between x and 0 = the RED segment on the number lines below.
|y| = the distance between y and 0 = the BLUE segment on the number lines below.
|x-y| = the distance BETWEEN X AND Y.

Statement 1: y<x
Case 1:

|x| - |y| = RED - BLUE.
|x-y| = RED - BLUE.
Thus, |x-y| = |x| - |y|.

Case 2:

|x| - |y| = RED - BLUE.
|x-y| = RED + BLUE.
Thus, |x-y| > |x| - |y|.
INSUFFICIENT.

Statement 2: xy<0
Since x and y have different signs, they are on OPPOSITE SIDES OF 0.

In each case:
|x| - |y| = RED - BLUE.
|x-y| = RED + BLUE.
Thus, |x-y| > |x| - |y|.
SUFFICIENT.

The correct answer is B.

Thanks GuruNY

Another approach:

|x - y| = the distance from x to y.

|x| = the distance from x to 0.

|y| = the distance from y to 0.

So we want to know

Is (distance from x to y) > (distance from x to 0) - (distance from y to 0) ?

S1::

If x > y, consider two options.

i: They're both positive. In this case, |x - y| = |x| - |y|, so the answer is NO.

ii: 0 > x and 0 > y. In this case, |x - y| is positive and |x| - |y| is negative, so the answer is YES.

Conflicting results, so insufficient.

S2::

One is positive, one is negative. Now the distance from x to y is the same as the distance from x to 0, then from 0 to y! In our equations, this can be written as

(the distance from x to y) = (the distance from x to 0) + (the distance from y to 0)

or

|x - y| = |x| + |y|

Since |x| + |y| must now be greater than |x| - |y|, this is SUFFICIENT.