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Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 16 Dec 2017, 00:05
TaN1213 wrote:
Bunuel wrote:
Basically the question asks whether the distance between the two points x and y on the line is greater than the difference between the individual distances of x and y from 0.

\(|x-y|>|x|-|y|\)?

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

A. ---------------\(0\)---\(y\)---\(x\)---, \(0<y<x\) --> in this case \(|x-y|>|x|-|y|\) becomes: \(x-y>x-y\) --> \(0>0\). Which is wrong;
B. ---------\(y\)---\(0\)---------\(x\)---, \(y<0<x\) --> in this case \(|x-y|>|x|-|y|\) becomes: \(x-y>x+y\) --> \(y<0\). Which is right, as we consider the range \(y<0<x\);
C. ---\(y\)---\(x\)---\(0\)--------------, \(y<x<0\) --> 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 \(|x-y|>|x|-|y|\):

A. ----\(y\)-----\(0\)-------\(x\)---, \(y<0<x\) --> in this case \(|x-y|>|x|-|y|\) becomes: \(x-y>x+y\) --> \(y<0\). Which is right, as we consider the range \(y<0<x\);
B. ----\(x\)-----\(0\)-------\(y\)---, \(x<0<y\) --> 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.

Answer: B.

Hi Bunuel,

Tried hard to understand the following related to the red part above but at the end no success. Would you please clear my doubt?

In 1 B & C, you have changed the sign of x and y in RHS, and not modified the LHS(red part) according to the sign of x and y.---- Understood the explanation of st. 1
But in 2 B, You have modified also the LHS part according to the sign of x and y. (red part) ------- Why have you not followed the process that you have followed in st 1 ?

What did this difference in procedure depend on?
Thank You in advance.


When \(x \le 0\) then \(|x|=-x\), or more generally when \(\text{some expression} \le 0\) then \(|\text{some expression}| = -(\text{some expression})\). For example: \(|-5|=5=-(-5)\);

When \(x \ge 0\) then \(|x|=x\), or more generally when \(\text{some expression} \ge 0\) then \(|\text{some expression}| = \text{some expression}\). For example: \(|5|=5\)



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

B. ----\(x\)-----\(0\)-------\(y\)---.

This means that \(x<0<y\). So, \(x - y < 0\), \(x<0\) and \(y>0\). According to the properties above if \(x - y < 0\), then \(|x-y|=-(x-y)\), if \(x<0\), then \(|x|=-x\) and if \(y>0\), then \(|y|=y\).

Hope it's clear.
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 16 Dec 2017, 10:56
Bunuel wrote:
When \(x \le 0\) then \(|x|=-x\), or more generally when \(\text{some expression} \le 0\) then \(|\text{some expression}| = -(\text{some expression})\). For example: \(|-5|=5=-(-5)\);

When \(x \ge 0\) then \(|x|=x\), or more generally when \(\text{some expression} \ge 0\) then \(|\text{some expression}| = \text{some expression}\). For example: \(|5|=5\)



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

B. ----\(x\)-----\(0\)-------\(y\)---.

This means that \(x<0<y\). So, \(x - y < 0\), \(x<0\) and \(y>0\). According to the properties above if \(x - y < 0\), then \(|x-y|=-(x-y)\), if \(x<0\), then \(|x|=-x\) and if \(y>0\), then \(|y|=y\).

Hope it's clear.

Following this same approach, shouldn't the following part become (-x-y)?
Quote:
1C. ---\(y\)---\(x\)---\(0\)--------------, \(y<x<0\) --> in this case \(|x-y|>|x|-|y|\) becomes: \(x-y>-x+y\) --> \(x>y\). Which is right, as we consider the range \(y<x<0\).


in 2 B you have modified the \(x-y\) to -(x-y) because x is negative and y is positive. So in the same way the 1C should take (-x-y) as per the sign of x.
What am I missing?
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 16 Dec 2017, 11:23
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TaN1213 wrote:
Bunuel wrote:
When \(x \le 0\) then \(|x|=-x\), or more generally when \(\text{some expression} \le 0\) then \(|\text{some expression}| = -(\text{some expression})\). For example: \(|-5|=5=-(-5)\);

When \(x \ge 0\) then \(|x|=x\), or more generally when \(\text{some expression} \ge 0\) then \(|\text{some expression}| = \text{some expression}\). For example: \(|5|=5\)



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

B. ----\(x\)-----\(0\)-------\(y\)---.

This means that \(x<0<y\). So, \(x - y < 0\), \(x<0\) and \(y>0\). According to the properties above if \(x - y < 0\), then \(|x-y|=-(x-y)\), if \(x<0\), then \(|x|=-x\) and if \(y>0\), then \(|y|=y\).

Hope it's clear.

Following this same approach, shouldn't the following part become (-x-y)?
Quote:
1C. ---\(y\)---\(x\)---\(0\)--------------, \(y<x<0\) --> in this case \(|x-y|>|x|-|y|\) becomes: \(x-y>-x+y\) --> \(x>y\). Which is right, as we consider the range \(y<x<0\).


in 2 B you have modified the \(x-y\) to -(x-y) because x is negative and y is positive. So in the same way the 1C should take (-x-y) as per the sign of x.
What am I missing?


C. ---\(y\)---\(x\)---\(0\)--------------, \(y<x<0\):

x - y > 0, so |x - y| = x - y;
y < 0, so |y| = y;
x < 0, so |x| = x.

\(|x-y|>|x|-|y|\) --> x - y > -x - (-y) --> x - y > -x + y
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 17 Dec 2017, 02:16
I am a little bit concerned here.
If the question states x and y, can we make an assumption that x ≠ y?
For example here, without this assumption, i can let x = y and (2) is NS?
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 17 Dec 2017, 02:22
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khiemchii wrote:
I am a little bit concerned here.
If the question states x and y, can we make an assumption that x ≠ y?
For example here, without this assumption, i can let x = y and (2) is NS?


Unless it is explicitly stated otherwise, different variables CAN represent the same number.

For (2) though x = y is not possible because in this case we'd get x^2 < 0, which is not true for any real number, so x = y does not satisfy xy < 0 and therefore should not be considered for the second statement. Similarly, for (1) it's give that y < x, so x = y cannot be true.
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 20 Jan 2018, 21:31
Can anyone tell me the flaw in my logic

|x - y| > |x| - |y|
Squaring both sides
x^2 +y^2-2xy> x^2 +y^2-2|x||y|
simplifying, xy<|x||y|,

It will only be possible when product of xy is -ve. or xy<0, Hence B.
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 22 Jan 2018, 03:27
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ManishKM1 wrote:
Can anyone tell me the flaw in my logic

|x - y| > |x| - |y|
Squaring both sides
x^2 +y^2-2xy> x^2 +y^2-2|x||y|
simplifying, xy<|x||y|,

It will only be possible when product of xy is -ve. or xy<0, Hence B.


Given:
|x - y| > |x| - |y|, you cannot square it. You can square only when you know that both sides are positive. Here, the right hand side may not be positive. For example, if x is 2 and y is 5.
3 > -3
Squaring does not work here since you get 9 > 9 which doesn't hold.
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 04 Mar 2018, 00:17
VeritasPrepKarishma

Is it incorrect to say that question is asking is |x|>|y| ?

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 05 Mar 2018, 05:23
Mudit27021988 wrote:
VeritasPrepKarishma

Is it incorrect to say that question is asking is |x|>|y| ?

Posted from my mobile device


For all real x and y, |x – y| >= |x| – |y|

|x – y| = |x| – |y| when
(1) x and y have the same sign and x has greater (or equal) absolute value than y
(2) y is 0

|x – y| > |x| – |y| in all other cases (i.e. y is not 0 and |x| < |y| or x and y have opposite signs)

So |x – y| > |x| – |y| is not the same as |x| > |y|

Check: https://www.veritasprep.com/blog/2014/0 ... -the-gmat/
https://www.veritasprep.com/blog/2014/0 ... t-part-ii/
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 12 Mar 2018, 17:51
Bunuel wrote:
Basically the question asks whether the distance between the two points x and y on the line is greater than the difference between the individual distances of x and y from 0.



Bunuel Xlnt interpretation of the QUESTION.
Personally feel if the above interpretation is understood and visualized on the number line the Question becomes easy to solve.

Thus the Question holds true if
1) Both x,y are +ve and ONLY if x<y (eg 3,4)
2) Both x,y are -ve and ONLY if x>y (eg -3,-4)
3) Both x,y are of OPPOSITE sign. Can be x>y OR x<y

Stat 1 y<x OR x>y
=> if both -ve, then 'Yes' from (2)
=> if both +ve then 'No' from (1)
Since no UNIQUE outcome so NOT SUFFICIENT

Stat 2 xy<0
=> so BOTH x,y are of OPPOSITE sign therefore ALWAYS 'Yes' from (3)
SUFFICIENT

Option B

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 26 Apr 2018, 09:03
VeritasPrepKarishma wrote:
Mudit27021988 wrote:
VeritasPrepKarishma

Is it incorrect to say that question is asking is |x|>|y| ?

Posted from my mobile device


For all real x and y, |x – y| >= |x| – |y|

|x – y| = |x| – |y| when
(1) x and y have the same sign and x has greater (or equal) absolute value than y
(2) y is 0

|x – y| > |x| – |y| in all other cases (i.e. y is not 0 and |x| < |y| or x and y have opposite signs)

So |x – y| > |x| – |y| is not the same as |x| > |y|

Check: https://www.veritasprep.com/blog/2014/0 ... -the-gmat/
https://www.veritasprep.com/blog/2014/0 ... t-part-ii/



Hi Karishma

I understand that either of the above mentioned two conditions should be satisfied to exclude the '=' from the inequality. In our case, the 1st condition is considered but how do we check that the absolute value of x is greater than y in this particular question.

Thank you for your help.
AS
Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0   [#permalink] 26 Apr 2018, 09:03

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