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

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Darkhorse12
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Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 07:00
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Is \(xy > 0\) ?


(1) \(|xy| + |x|y + x|y| + xy > 0\)

(2) \(-x < -y < |y|\)
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D
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rohit8865
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 07:48
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Darkhorse12 wrote:
Is xy>0?
1. |xy| + |x|y + x|y| + xy > 0
2. -x< -y < |y|


Anybody can help how to approach this question?


1. take 4 cases
a) y>0 , x<0 then |xy| + |x|y + x|y| + xy =0 but it should be >0 as condition given in option....thus y>0 , x<0 is not the case
b) y<0 , x>0 same as above
c)y<0 , x<0 same as above
d) y>0 , x>0 then |xy| + |x|y + x|y| + xy = 4xy >0 .....thus x>0 and y>0 is only satisfying
thus suff

2. givn condition only prevails when y>0 and thus x is also >0
suff

Ans D
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 07:50
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Darkhorse12 wrote:
Is xy>0?
1. |xy| + |x|y + x|y| + xy > 0
2. -x< -y < |y|


Anybody can help how to approach this question?


Hi,
Anything that tells us what are the signs of x and y will be sufficient..
If x and y are SAME sign, ans is YES...
If x and y are OPPOSITE, ans is NO..

Let's see the statements..
1. |xy| + |x|y + x|y| + xy > 0
If only x<0, |xy| and |x|y will be positive and x|y| and xy will be NEGATIVE..
When you add all four terms and will be 0..
Similarly for only y<0..
Thus both are of same sign..
Sufficient

2. -x< -y < |y|
In -y<|y|, y will be positive otherwise both would have been EQUAL..
-X<-y means x is positive, which makes -x as negative..
Again xy>0..
Sufficient

D
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 09:37
chetan2u wrote:
Darkhorse12 wrote:
Is xy>0?
1. |xy| + |x|y + x|y| + xy > 0
2. -x< -y < |y|


Anybody can help how to approach this question?


Hi,
Anything that tells us what are the signs of x and y will be sufficient..
If x and y are SAME sign, ans is YES...
If x and y are OPPOSITE, ans is NO..

Let's see the statements..
1. |xy| + |x|y + x|y| + xy > 0
If only x<0, |xy| and |x|y will be positive and x|y| and xy will be NEGATIVE..
When you add all four terms and will be 0..
Similarly for only y<0..
Thus both are of same sign..
Sufficient

2. -x< -y < |y|
In -y<|y|, y will be positive otherwise both would have been EQUAL..
-X<-y means x is positive, which makes -x as negative..
Again xy>0..
Sufficient

D




Hello chetan2u,

In statement-1 as you mentioned
If only x<0, |xy| and |x|y will be positive and x|y| and xy will be NEGATIVE..
When you add all four terms and will be 0..
Similarly for only y<0..

Also if both are x and y are negative then also the result is 0.

Its only true when x and y both are positive.


Hence Sufficient. Right??
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 09:59
#1 is equivalent to (|x|+x)*(|y|+y)>0

Now, the terms (|z|+z) can only be positive or zero. As the inequality is stricly greater than zero, the value of zero for any term in the parenthesis is not allowed. Then necessarily (|z|+z)>0 (strictly) for each z=x and z=y. This in turn means that x>0 and y>0 (strictly). Therefore, xy>0 (sufficient)

#2 Can be divided in two: -x<-y and -y<|y|
Restating: (i) x>y and (ii) |y|+y>0
As we have seen (ii) means y>0, and then due to (i), x>0. Therefore, xy>0 (sufficient)

So, the answer is D
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 10:37
Darkhorse12 wrote:
Also if both are x and y are negative then also the result is 0. .... Right??


Yes Darkhorse12,

If both are negative then

|xy| --> +xy

xy --> +xy

x|y| --> -xy

|x|y --> -xy

xy + xy - xy - xy = 0

also x=y=0 is also not valid.

so both x & y has to be +ve.
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 11:04
chetan2u wrote:
Darkhorse12 wrote:
Is xy>0?
1. |xy| + |x|y + x|y| + xy > 0
2. -x< -y < |y|


Anybody can help how to approach this question?


Hi,
Anything that tells us what are the signs of x and y will be sufficient..
If x and y are SAME sign, ans is YES...
If x and y are OPPOSITE, ans is NO..

Let's see the statements..
1. |xy| + |x|y + x|y| + xy > 0
If only x<0, |xy| and |x|y will be positive and x|y| and xy will be NEGATIVE..
When you add all four terms and will be 0..
Similarly for only y<0..
Thus both are of same sign..
Sufficient

2. -x< -y < |y|
In -y<|y|, y will be positive otherwise both would have been EQUAL..
-X<-y means x is positive, which makes -x as negative..
Again xy>0..
Sufficient

D



Nice explanation . Thanks !!
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 11:16
seems harder. By the way, what is the exact source of this question?
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 18:27
Expert Reply
Darkhorse12 wrote:
chetan2u wrote:
Darkhorse12 wrote:
Is xy>0?
1. |xy| + |x|y + x|y| + xy > 0
2. -x< -y < |y|


Anybody can help how to approach this question?


Hi,
Anything that tells us what are the signs of x and y will be sufficient..
If x and y are SAME sign, ans is YES...
If x and y are OPPOSITE, ans is NO..

Let's see the statements..
1. |xy| + |x|y + x|y| + xy > 0
If only x<0, |xy| and |x|y will be positive and x|y| and xy will be NEGATIVE..
When you add all four terms and will be 0..
Similarly for only y<0..
Thus both are of same sign..
Sufficient

2. -x< -y < |y|
In -y<|y|, y will be positive otherwise both would have been EQUAL..
-X<-y means x is positive, which makes -x as negative..
Again xy>0..
Sufficient

D




Hello chetan2u,

In statement-1 as you mentioned
If only x<0, |xy| and |x|y will be positive and x|y| and xy will be NEGATIVE..
When you add all four terms and will be 0..
Similarly for only y<0..

Also if both are x and y are negative then also the result is 0.

Its only true when x and y both are positive.


Hence Sufficient. Right??


Yes ...

We are looking for an answer that tells us that x and y are of same sign..
Statement I tells us that both x and y are both POSITIVE hence sufficient..
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   20 Feb 2017, 22:27
Darkhorse12 wrote:
Is xy>0?
1. |xy| + |x|y + x|y| + xy > 0
2. -x< -y < |y|


Anybody can help how to approach this question?

My reasoning:

A- lets take 4 possibility ..
x,y = (+,+) ,(-,+) , (+,-) , (-,-)
1st case (+,+) -- overall we will get positive , so fine
2nd case -(-,+) -- we will get 0 --discard
3rd case --same as second
4th case --we will get a negative --discard ...
only viable possibility is x any can be positive..
suff

B--|Y| always be +ve...
-Y < |Y| can be only possible when Y is positive ...
-X < -Y
RHS is negative (Proved above).
Value of X have to be +ve ..else x < -Y (wrong)
Suff ..

So D...
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink] New post   24 Jan 2023, 07:45
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Re: Is xy > 0 ? (1) |xy| + |x|y + x|y| + xy > 0 (2) -x < -y < |y| [#permalink]
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