Is |x - y| > |x| - |y|?
(1) y < x
(2) xy < 0
OA=B. Could we solve by algebra?
Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0
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Hi ziyuenlau,ziyuenlau wrote:Is |x - y| > |x| - |y|?
(1) y < x
(2) xy < 0
OA=B. Could we solve by algebra?
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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The first part yes, the second part no. It's also true if y > x > 0 or if 0 > x > y. For example,Jay@ManhattanReview wrote:
The inequality |x - y| > |x| - |y| is true if x and y have the opposite signs, else |x - y| = |x| - |y|.
|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|.
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Yes, Matt, you are right. Thank you.Matt@VeritasPrep wrote:The first part yes, the second part no. It's also true if y > x > 0 or if 0 > x > y. For example,Jay@ManhattanReview wrote:
The inequality |x - y| > |x| - |y| is true if x and y have the opposite signs, else |x - y| = |x| - |y|.
|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|.
-Jay
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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,
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An efficient approach is to plot the distances on a NUMBER LINE.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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Solution:hazelnut01 wrote: ↑Tue Apr 25, 2017 12:31 amIs |x - y| > |x| - |y|?
(1) y < x
(2) xy < 0
OA=B. Could we solve by algebra?
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
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