Question: Is |x - y| > |x| - |y|?

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

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

“Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:
Approach Solution 1:
Probably the best way to solve this problem is the plug-in method. Though there are two properties worth to remember:

1. Always true:|x+y|≤|x|+|y|, note that "=" sign holds for xy ≥ 0 (or simply when x and y have the same sign);
2. Always true: |x−y|≥|x|−|y|, note that "=" sign holds for xy > 0 (so when x and y have the same sign) and |x|>|y| (simultaneously). (Our case)

So, the question basically asks whether we can exclude "=" scenario from the second property.

(1) y < x --> we can not determine the signs of x and y. Not sufficient.
(2) xy < 0 --> "=" scenario is excluded from the second property, thus |x−y|>|x|−|y|. Sufficient.

Correct Answer: B

Approach Solution 2:
|x−y|>|x|−|y|?

(1) y < x, 3 possible cases for |x−y|>|x|−|y|:

1. ---------------0---y---x---, 0 < y < x --> in this case |x−y| > |x|−|y| becomes: x−y > x−y --> 0>0. Which is wrong;
2. ---------y---0---------x---, y < 0 < x --> in this case |x−y| > |x|−|y| becomes: x−y > x+y --> y<0. Which is right, as we consider the range y < 0 < x;
3. ---y---x---0--------------, y<x<0 --> in this case |x−y| >| x|−|y| becomes: x−y > −x+y --> x > y. Which is right, as we consider the range y < x < 0.

Two different answers. Not sufficient.

(2) xy < 0, means x and y have different signs, hence 2 cases for |x−y| > |x|−|y|:

1. ----y-----0-------x---, y < 0 < x --> in this case |x−y| > |x|−|y| becomes: x−y > x+y --> y < 0. Which is right, as we consider the range y < 0 < x;
2. ----x-----0-------y---, x < 0 < y --> in this case |x−y| > |x|−|y| becomes: −x+y > −x−y --> y>0. Which is right, as we consider the range x < 0 < y.

In both cases inequality holds true. Sufficient.

Correct Answer: B

Approach Solution 3:
The inequality |x - y| > |x| - |y| is true if x and y have the opposite signs, else |x - y| = |x| - |y|.

Example:

1. Say x = 3 and y = 2, then |x - y| = |3-2| = 1 AND |x| - |y| = |3| - |2| = 1 => |x - y| = |x| - |y|.
2. Say x = -3 and y = -2, then |x - y| = |-3 + 2| = 1 AND |x| - |y| = |-3| - |-2| = 3 - 2 = 1 => |x - y| = |x| - |y|.
3. Say x = 3 and y = -2, then |x - y| = |3 - (-2)| = |3+2| = 5 AND |x| - |y| = |3| - |-2| = 3 - 2 = 1 => |x - y| > |x| - |y|.
4. Say x = -3 and y = 2, then |x - y| = |(-3) -2| = |-3-2| = 5 AND |x| - |y| = |-3| - |2| = 3 - 2 = 1 => |x - y| > |x| - |y|.

So, in a nutshell, we have to see if x and y have the same or the opposite signs.

Statement 1: y < x

The inequality y < x can hold true if x and y have the same or the opposite signs.

Example: 2 < 3 and -2 < 3. Insufficient.

Statement 1: xy < 0

The inequality xy < 0 implies that xy is negative and this is possible if x and y have the opposite sign. Sufficient.

Correct Answer: B

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