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

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Mon Jan 15, 2018 9:00 pm
VJesus12 wrote:Is |x - y | > |x | - |y | ?

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

The OA is the option B.

I don't know how to solve this DS question.

Experts, may you give me some help? Please.
We have to find out if |x - y | > |x | - |y |.

(1) y < x

Case 1: x = 2 and y = 1, then |x - y | > |x | - |y | => |2 - 1 | ? |2 | - |1 | => 1 = 1. The answer is No.
Case 2: x = 2 and y = -1, then |x - y | > |x | - |y | => |2 + 1 | ? |2 | - |-1 | => 3 ? 2 - 1 => 3 > 1. The answer is Yes. No unique answer. insufficient.

(2) xy < 0

=> One between x and y is positive and the other is negative.

Case 1: x is positive and y is negative.

|x - y | > |x | - |y | => x + |y | ? |x | - |-y | => x + |y | ? |x | - |y | => Positive + Positive > Positive - Positive. The answer is Yes.

Case 2: y is positive and x is negative.

|x - y | > |x | - |y | => |-x - y | ? |-x | - |y | => |x + y | ? |x | - |y | => Positive + Positive > Positive - Positive. The answer is Yes. A unique answer. Sufficient.

The correct answer: B

Hope this helps!

-Jay
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by Scott@TargetTestPrep » Thu Jan 18, 2018 1:50 pm
VJesus12 wrote:Is |x - y | > |x | - |y | ?

(1) y < x
(2) xy < 0
Statement One Alone:

y < x

Let's pick some convenient numbers for x and y.

1) If y = 1 and x = 2, we have:

|2 - 1| > |2| - |1| ?

1 > 1? The answer is "No."

2) If y = -1 and x = 2, we have:

|2 - (-1)| > |2| - |-1| ?

3 > 1? The answer is "Yes."

We can see that statement one alone is not sufficient to answer the question.

Statement Two Alone:

xy < 0

Since xy < 0, x cannot equal y. Then, there are two cases: Either x > y or y > x.

Case 1: x > y

Since x and y have opposite signs and since x > y, it must be true that x > 0 and y < 0. Then, the given inequality becomes:

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

Is x - y > x + y ?

Is -y > y?

Is 0 > 2y ?

Is y < 0 ?

Since y is negative in this case, all the above inequalities hold. Thus, |x - y| is greater than |x| - |y|.

Case 2: x < y

Since x and y have opposite signs, it must be true that x < 0 and y > 0. Then, the given inequality becomes:

Is -x + y > -x - y? (Notice that x - y < 0; therefore |x - y| = -(x - y))

Is y > - y?

Is 2y > 0?

Is y > 0?

Since y is positive in this case, all the above inequalities hold. Thus, |x - y| is, once again, greater than |x| - |y|.

Answer: B

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by GMATGuruNY » Fri Jan 19, 2018 5:29 am
VJesus12 wrote:Is |x - y | > |x | - |y | ?

(1) y < x
(2) xy < 0
|a-b| = the distance between a and b.
|a| = the distance between a and 0.

Statement 2: xy < 0
Since x and y have different signs, they are to opposite sides of 0:
Case 1:
x<----|x|---->0<----|y|---->y
Case 2:
y<----|y|---->0<----|x|---->x.

In each case, the distance between x and y is equal to the sum of the blue distance and the red distance:
|x-y| = |x| + |y|.

Substituting |x-y| = |x| + |y| into |x-y| > |x| - |y|, we get:
Is |x| + |y| > |x| - |y|?
Subtracting |x| from both sides, we get:
Is |y| > -|y|?
Since the left side is POSITIVE, while the right side is NEGATIVE, the answer to the rephrased question stem is YES.
SUFFICIENT.

Statement 1: y < x
Statement 1 is satisfied by Case 2, in which y<0 and x>0.
In Case 2, the answer to the question stem is YES.

Case 3: y=1 and x=2
In this case:
|x-y| = |2 -1| = 1.
|x| - |y| = |2| - |1| = 1.
Since |x-y| = |x| - |y|, the answer to the question stem is NO.

Since the answer is YES in Case 2 but NO in Case 3, INSUFFICIENT.

The correct answer is B.
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by GMATGuruNY » Sat Jan 20, 2018 3:55 am
VJesus12 wrote:Is |x - y | > |x | - |y | ?

(2) xy < 0
I received a PM requesting that I evaluate Statement 2 algebraically.

|a-b| = the NONEGATIVE difference between a and b.
Thus:
If a > b, then |a-b| = a-b.
If a < b, then |a-b| = b-a.

Also:
|a| = a when a>0.
|a| = -a when a<0.

Statement 2: xy < 0
Case 1: x>0 and y<0
In this case:
|x-y| = x-y
|x| = x
|y| = -y.

Substituting the blue expressions into the question stem, we get:
Is x-y > x - (-y) ?

Simplifying the question stem in red, we get:
x-y > x - (-y)
-y > y
0 > 2y
0 > y
Is y < 0 ?
Since Case 1 is based on the condition that y<0, the answer is YES.

Case 2: x<0 and y>0
In this case:
|x-y| = y-x
|x| = -x
|y| = y.

Substituting the blue expressions into the question stem, we get:
Is y-x > -x - y ?

Simplifying the question stem in red, we get:
y-x > -x - y
y > -y
2y > 0
Is y > 0 ?
Since Case 2 is based on the condition that y>0, the answer is YES.

Since the answer is YES in both cases, SUFFICIENT.
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