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

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Math Expert
Joined: 02 Sep 2009
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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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 $$y<x<0$$.

(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.

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?

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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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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Posts: 44665
Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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16 Dec 2017, 11:23
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Expert's post
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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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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17 Dec 2017, 02:22
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Expert's post
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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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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22 Jan 2018, 03:27
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Expert's post
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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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 29 Nov 2016 Posts: 52 Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink] Show Tags 04 Mar 2018, 00:17 VeritasPrepKarishma Is it incorrect to say that question is asking is |x|>|y| ? Posted from my mobile device Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8025 Location: Pune, India Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink] Show Tags 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/ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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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

Regards
Dinesh
Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0   [#permalink] 12 Mar 2018, 17:51

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