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Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y

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Re: Hard inequality: Is xy < 0  [#permalink]

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New post 07 Nov 2013, 05:33
Temurkhon wrote:
Bunuel wrote:
Is \(xy<0\)?

(1) \(\frac{x^3*y^5}{x*y^2}<0\)

(2) \(|x|-|y|<|x-y|\)

Yes, OA is E.

Number plugging as AKProdigy87 did is probably the best way here. But still if needed below is short overview of algebra of this question:

(1) y<0. Not sufficient.

(2) Holds true when:
A. 0<x<y
B. x<0<y
C. y<x<0
D. y<0<x

Not sufficient.

(1)+(2) C. and D. options are left from (2) but still insufficient, as xy may or may not be negative.


Answer: E.


I agree. Inequality rule saying that |x|-|y|<|x-y| is true only when x and y are having different signs exists. Does not it? If so, B is correct


The answer is not B. Consider the following cases to discard: x=-1 and y=-2, for a NO answer and x=1 and y=-2 for an YES answer.

The property you are referring to says that inequality \(|x-y|\geq{|x|-|y|}\) holds true for any values of x and y. How are you applying it to this problem?
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Re: Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y  [#permalink]

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New post 09 Nov 2013, 09:10
Sorry Bunuel. I realized my mistake. I am clear now!
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Re: Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y  [#permalink]

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New post 11 Mar 2014, 03:04
Bunuel wrote:
Is \(xy<0\)?

(1) \(\frac{x^3*y^5}{x*y^2}<0\)

(2) \(|x|-|y|<|x-y|\)


Hi Bunuel, I did it like this.

1) \(\frac{x^3*y^5}{x*y^2}<0\)

here, we can get two cases
a) x = +ve and y = -ve
this satisfies xy < 0
b) x = -ve and y = -ve
this does not satisfy xy < 0
so, 1 alone is insufficient.

2) \(|x|-|y|<|x-y|\)

here also we would get the same 2 cases.
so, 2 alone is insufficient.

Also, even if we combine both the fact statements they would be insufficient.
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Re: Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y  [#permalink]

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New post 02 May 2016, 04:03
Hi Bunuel
please help me in how to deal with the following expression: |x|−|y|<|x−y| in DS questions in simple way
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Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y  [#permalink]

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New post 02 May 2016, 04:32
hatemnag wrote:
Hi Bunuel
please help me in how to deal with the following expression: |x|−|y|<|x−y| in DS questions in simple way


Please read the entire thread to understand as it has been discussed at is-xy-0-1-x-3-y-5-x-y-2-0-2-x-y-x-y-86780.html#p651477 and is-xy-0-1-x-3-y-5-x-y-2-0-2-x-y-x-y-86780.html#p651854

As for the theory or understanding behind the expression: |x|-|y| < |x-y|, you need to understand what do |x| and |x-y| mean in isolation.

|x| is the distance of 'x' from absolute 0 on the number line.

|x-y| represents the distance of 'x' from 'y' on the number line.

Thus, for S2, you are told that |x|-|y| < |x-y| ---> for such 'complex' absolute value inequality expressions, use numbers to guide you but for algebraic treatment, read along:

|x|-|y| < |x-y| ---> distance of x - distance of y < distance of x from y

Also, note that |any value| = always non-negative ---> |any number| \(\geq\) 0

Thus, |x| -|y| < a non negative quantity ---> |x| - |y| < 0 (I took 0 as it the simplest and the most straightforward example of a non-negative number).

2 cases on the number line that follow the fact that |x| - |y| < 0 :

Attachment:
5-02-16 8-28-52 AM.jpg
5-02-16 8-28-52 AM.jpg [ 32.91 KiB | Viewed 733 times ]


For case 1, you get xy<0 but for case 2, you get xy> 0 ---> S2 is NOT sufficient as it is giving us 2 different answers to the same question asked.

Hope this helps.
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Re: Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y  [#permalink]

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New post 02 May 2016, 12:29
Thank you so much Engr2012
Actually, i know the basic concepts of Absolute value, but such expressions make me confused in DS question and i feel disable to analyze it.
I see that using the number line is very helpful to study the two cases of such expression.
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Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y  [#permalink]

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New post 20 Jan 2017, 07:02
Yes Statement 2 is correct if you check it for x >0 and y<0 and x<0 and y>0 (i.e when x and y have opposite signs) in which case xy<0.
However, Statement 2 is true when you x>0 and y>0 too in which case xy is not less than 0.
Hence, Statement 2 is not sufficient to determine if xy<0. Hence, B is not the correct option.

Now, when you combine Statement 1 and 2, you get that y is definitely <0. You need to now determine the sign of x. But we see that Statement 2 is true for both x>0 and x<0 when y<0. Hence,even the combined statements are not sufficient to determine the unique sign of x which is important to find if xy <0 for sure.

Hence, answer is E.
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Re: Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y  [#permalink]

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Re: Is xy<0 ? (1) x^3*y^5/x*y^2 <0 (2) |x|-|y|<|x-y &nbs [#permalink] 16 Aug 2018, 03:47

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