Saffa wrote:I've searched the forums, but due to the ABS or | | notation I could not find it.

Is ABS(x - y) > ABS(x) - ABS(y)?

(1) y < x
(2) xy < 0

OA is B

For (2) I use substitution

x y
-6 3
-1 1
1 -1
6 -3

And for all these combination it worked. Is there is shorter way?

There is, but it's more conceptual than choosing numbers. If you understand that absolute value measures distance, these kinds of questions can normally be done quickly and reliably:

|x| is the distance from x to zero
|x-y| is the distance between x and y on the number line

Consider the inequality in question:

|x - y| > |x| - |y|


Take Statement 2 first. It tells us that x and y are on opposite sides of zero. Draw this on a number line (and it really makes no difference whether y or x is on the left):

---y----0-----x-----

Clearly the distance from x to y is equal to the distance from x to 0 plus the distance from y to zero. That is,

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

which must be greater than |x| - |y|.

Statement 1 is not sufficient; we might have the situation above, but we might also have:

------0----y-----x----

Here, |x-y| is exactly equal to |x| - |y| --- the distance from x to y is equal to the distance from x to zero minus the distance from y to zero.

That may take some time to explain, but when you get accustomed to looking at absolute value problems in this way, it can be very fast to do.