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.
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Inequalities
Source: Beat The GMAT — Data Sufficiency |
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An efficient approach is to plot the distances on a NUMBER LINE.psm12se wrote:Is |x-y| > |x| - |y| ?
1) y < x
2) xy < 0
|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.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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