Data Sufficiency

ziyuenlau wrote:Is |x - y| > |x| - |y|?

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

OA=B. Could we solve by algebra?

Hi ziyuenlau,

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.

The correct answer: B

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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Matt@VeritasPrep wrote:

Jay@ManhattanReview wrote:
The inequality |x - y| > |x| - |y| is true if x and y have the opposite signs, else |x - y| = |x| - |y|.

The first part yes, the second part no. It's also true if y > x > 0 or if 0 > x > y. For example,

|3 - 5| > |3| - |5|

and

|-3 - (-5)| > |-3| - |-5|

This follows from |y| > |x| leading to 0 > |x| - |y| and the fact that |x - y| ≥ 0 for any x and y.

Combining those two inequalities gives us |x - y| > |x| - |y|, so this will always hold if |y| > |x|.

Yes, Matt, you are right. Thank you.

-Jay
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Manhattan Review GMAT Prep

Locations: New York | Beijing | Auckland | Milan | and many more...

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hazelnut01 wrote:Is |x - y| > |x| - |y|?

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

OA=B. Could we solve by algebra?

Target question: Is |x - y| > |x| - |y|?

Statement 1: y < x
Let's test some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 2 and y = 1. In this case, |x - y| = |2 - 1| = 1 and |x| - |y| = |2| - |1| = 1. So, the answer to the target question is NO, |x - y| is NOT greater than |x| - |y|
Case b: x = 2 and y = -1. In this case, |x - y| = |2 - (-1)| = 3 and |x| - |y| = |2| - |-1| = 1. So, the answer to the target question is YES, |x - y| IS greater than |x| - |y|
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: xy < 0
This tells us that one value (x or y) is positive, and the other value is negative. This sets up two possible cases:

Case a: x is positive and y is negative.
So, we're taking a positive value (x) and subtracting a negative value (y). Doing so yields a positive value that is bigger than x.
In other words, we have: 0 < x < |x - y|

Now let's examine |x| - |y|
Since x is positive, we know that |x| = x
Since y ≠ 0, we know that 0 < |y|
So, |x| - |y| = x - |y| = some number less than x
In other words, |x| - |y| < x

When we combine the inequalities we get: |x| - |y| < x < |x - y|
In this case, the answer to the target question is YES, |x - y| IS greater than |x| - |y|


Case b: x is negative and y is positive.
Here, we're taking a negative value (x) and subtracting a positive value (y). Doing so yields a negative value that is less than x.
In other words, we have: x - y < x < 0 < |x|
Important: since the MAGNITUDE of x - y is greater than the MAGNITUDE of x, we can write: |x| < |x - y|

Now let's examine |x| - |y|
Since x ≠ 0, we know that 0 < |x|
Since |x| is a positive number, we know that subtracting |y| (another positive value) will yield a number that is LESS THAN |x|
In other word, |x| - |y| < |x|

When we combine the inequalities we get: |x| - |y| < |x| < |x - y|
In this case, the answer to the target question is YES, |x - y| IS greater than |x| - |y|

In both possible cases, the answer to the target question is the same: YES, |x - y| IS greater than |x| - |y|
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: B

Cheers,
Brent

hazelnut01 wrote: ↑

Tue Apr 25, 2017 12:31 am

Is |x - y| > |x| - |y|?

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

OA=B. Could we solve by algebra?

Solution:

Notice that x = (x - y) + y; thus, using the triangle inequality, we have:

|x| = |(x - y) + y| ≤ |x - y| + |y|

|x - y| ≥ |x| - |y|

This tells me that |x - y| can be greater than |x| - |y| or can be equal to |x| - |y|, but it can never be less than |x| - |y|. So, if I look for counter examples, I won’t waste time trying to look for numbers where |x - y| is less than |x| - |y|.

Statement One Alone:

If x > y, then x - y is positive; hence the given question reduces to “Is x - y > |x| - |y|?”. We immediately notice that if x and y are both positive, then |x| - |y| is equal to x - y. In this case, the answer to the question is no. We can, if we want, verify by letting x = 2 and y = 1.

If x and y are not both positive (which is only possible when x is positive but y is negative), then we have: “Is x - y > x - (-y)?” The inequality simplifies to “Is -y > y?”. Since y is negative, the answer to this question is yes. Again we can verify by taking values such as x = 2 and y = -1.

Statement one alone is not sufficient.

Statement Two Alone:

Notice that the given information in statement two tells us that neither x nor y is zero, x and y are not equal and they have opposite signs. All we have to do is to consider the cases when x is positive and x is negative to find the answer.

If x is positive, then y is negative. Thus, the expression x - y is positive, and the question becomes:

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

Is -y > y?

Since y is negative, the answer is yes.

If x is negative, then y is positive. Thus, the expression x - y is negative, and the question becomes:

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

Is -x + y > -x - y?

Is y > -y?

Since y is positive, the answer is again yes. We see that |x - y| > |x| - |y| is satisfied whenever xy < 0.

Thus, statement two is sufficient to answer the question.

Answer: B