Inequalities

[psm12se]

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

[ceilidh.erickson]

When you're looking at absolute value questions with inequalities, you need to think about positives and negatives (absolute value is the positive distance from 0). So ask yourself: when would |x - y| be greater than |x| - |y|?

If x and y are equal, |x - y| and |x| - |y| will always be equal. |3 - 3| = |3| - |3|

If x and y are both positive, these could be the same. Try x = 3 and y = 2: |3 - 2| = |3| - |2| -> 1 = 1
The first could be greater, though, if |x| < |y|. Try x = 2 and y = 3: |2 - 3| > |2| - |-3| -> 1 > -1

If x and y are both negative, again these could be the same. Try x = -3 and y = -2: |-3 - (-2)| = |-3| - |-2| -> 1 = 1
The first could be greater, though. Try x = -2 and y = -3: |-2 - (-3)| > |-2| - |-3| -> 1 > -1

If x and y have different signs, then |x - y| will always be greater than |x| - |y|
x = 3 and y = -2: |3 - (-2)| > |3| - |-2| -> 5 > 1
x = -3 and y = 2: |-3 - 2| > |-3| - |2| -> 5 > 1
x = 2 and y = -3: |2 - (-3)| > |2| - |-3| -> 5 > -1
x = -2 and y = 3: |-2 - 3| > |-2| - |3| -> 5 > -1

So, what we really need to know is... do x and y have the same sign? If the signs are different, it's sufficient. If the signs are the same, we need to know whether |x| > |y|

Statement (1) tells us that y is less than x, but tells us nothing about the signs or the relative distances from 0. Insufficient.

Statement (2) tells us that x and y have different signs. We don't know which one is positive and which one is negative, but it doesn't matter; we know that |x - y| will be greater than |x| - |y|. Sufficient.

The answer is B.

[psm12se]

Thanks for the response, answer is indeed : B

[GMATGuruNY]

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

An efficient 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.