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Math Expert V
Joined: 02 Sep 2009
Posts: 55804
If x ≠ 0, is xy > 0?  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 64% (01:13) correct 36% (01:12) wrong based on 188 sessions

HideShow timer Statistics If x ≠ 0, is xy > 0?

(1) x > 0
(2) 1/x < y

Kudos for a correct solution.

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Math Expert V
Joined: 02 Aug 2009
Posts: 7765
Re: If x ≠ 0, is xy > 0?  [#permalink]

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Bunuel wrote:
If x ≠ 0, is xy > 0?

(1) x > 0
(2) 1/x < y

Kudos for a correct solution.

The Q asks xy>0.. which basically means :"are x and y of the same sign?"
1)statement 1 tell us that x>0.. nothing about y.. if y is +ive, then yes .. if y is -ive, ans is no... insuff
2) statement two also does not give us a clear answer.. it can take both values as negative , both positive or x -ive and y positive... insuff
combined it gives us x >0.. by statement 2 y will be positive for positive x.... suff
C
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Re: If x ≠ 0, is xy > 0?  [#permalink]

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1
Bunuel wrote:
If x ≠ 0, is xy > 0?

(1) x > 0
(2) 1/x < y

Kudos for a correct solution.

1) x >0 but Y can be > 0 or Y can be < 0 . So not sufficient.
2) Y> 1/x here Y can be +ve and X can be -ve or both X and Y can be +ve . So not sufficient .

Combining 1+2
Since X>0 we can multiply by X on both sides and sign of inequality will not change
=> 1 < XY thus XY >0 so C
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Re: If x ≠ 0, is xy > 0?  [#permalink]

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1
Bunuel wrote:
If x ≠ 0, is xy > 0?

(1) x > 0
(2) 1/x < y

Kudos for a correct solution.

Hello,

For xy >0 both x an y should be of the same sign either both positive or both negative. Our job is to find whether both carries same sign or opposite.

Statement1 : X is positive. No information on Y, it can take both positive and negative sign hence not sufficient
Statement 2 : 1/x<y . In this equation x can be positive making 1/x +ive or X can be negative making 1/x -ive. Y is greater than 1/x hence it can be both positive and negative.

Combining both statements are sufficient.
From statement 1 we get to know that x is postive and hence 1/x is +ive which results in Y being positive.
From both statements we know that both x and y are positive so xy>0. Sufficient
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Re: If x ≠ 0, is xy > 0?  [#permalink]

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from statement 2 1/x<y
y>1/x

y-1/x>0

xy-1/x>0

xy-1>0

xy>1.

So statement 2 itself sufficient.
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Re: If x ≠ 0, is xy > 0?  [#permalink]

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It's no where given x is positive. You can't simply get rid of x in the second last equation. If x is positive then you're right but if x is -ive then the sign will flip. Hope I am able to articulate my point.
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Re: If x ≠ 0, is xy > 0?  [#permalink]

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Bunuel wrote:
If x ≠ 0, is xy > 0?

(1) x > 0
(2) 1/x < y

Kudos for a correct solution.

Question : Is xy > 0?

for xy to be greater than zero, both x and y must have same sign so the question can be rephrased as

Question : Do x and y have same sign?

Statement 1: x > 0

But this doesn't provide any information of y nor does it provide any relation of x with y

hence, NOT SUFFICIENT

Statement 2: 1/x < y

i.e. If y is Negative, x Must be negative because 1/x will be defined as less than negative value (i.e. y)
and If y is Positive, x May be Positive or negative because 1/x will be defined as less than Positive value (i.e. y) e.g. x=1, y=2 OR x=-1, y=2

hence, NOT SUFFICIENT

Combining the two statements

x>0 and 1/x < y
for x>0, 1/x MUST be Positive
and Since, 1/x(positive value) is lass than y, therefore y must be a POSITIVE value as well

hence, SUFFICIENT

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Re: If x ≠ 0, is xy > 0?  [#permalink]

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Psiva00734 wrote:
from statement 2 1/x<y
y>1/x

y-1/x>0

xy-1/x>0

xy-1>0

xy>1.

So statement 2 itself sufficient.

There are two highlighted steps above, The Second Highlighted step after the first highlighted step is incorrect due to the reason that sign of x is unknown so you can't multiple x on both sides and keep the sign same.

Sign in the inequation after multiplying x on both sides may or may not be same depending on the sign of x being positive or negative respectively.

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Math Expert V
Joined: 02 Sep 2009
Posts: 55804
Re: If x ≠ 0, is xy > 0?  [#permalink]

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Bunuel wrote:
If x ≠ 0, is xy > 0?

(1) x > 0
(2) 1/x < y

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

(1) INSUFFICIENT: This tells us nothing about the sign of y.

In evaluating Statement (2), you might be tempted to assume that x must be positive. After all, we just read information in Statement (1) that tells us that x is positive. Besides, it is natural to assume that a given variable will have a positive value, because positive numbers are much more intuitive than negative numbers.

Instead, if we follow Principle #4, we will actively try to violate Statement (1), helping us expose the trick in this question.

(2) INSUFFICIENT: If we contradict Statement (1) to consider the possibility that x is negative, we would realize that it is necessary to flip the sign of the inequality when we cross multiply. That is, if x < 0, then 1/x < y means that 1 > xy, and the answer to the question is MAYBE.

(1) & (2) SUFFICIENT: If x is positive, then statement (2) says that 1 < xy (we do not flip the sign when cross multiplying). Thus, xy > 0.

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Re: If x ≠ 0, is xy > 0?  [#permalink]

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I dint get this - How combined statements are sufficient? Could someone explain by inserting values ?
Manager  B
Joined: 30 Apr 2013
Posts: 77
Re: If x ≠ 0, is xy > 0?  [#permalink]

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Bunuel wrote:
Bunuel wrote:
If x ≠ 0, is xy > 0?

(1) x > 0
(2) 1/x < y

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

(1) INSUFFICIENT: This tells us nothing about the sign of y.

In evaluating Statement (2), you might be tempted to assume that x must be positive. After all, we just read information in Statement (1) that tells us that x is positive. Besides, it is natural to assume that a given variable will have a positive value, because positive numbers are much more intuitive than negative numbers.

Instead, if we follow Principle #4, we will actively try to violate Statement (1), helping us expose the trick in this question.

(2) INSUFFICIENT: If we contradict Statement (1) to consider the possibility that x is negative, we would realize that it is necessary to flip the sign of the inequality when we cross multiply. That is, if x < 0, then 1/x < y means that 1 > xy, and the answer to the question is MAYBE.

(1) & (2) SUFFICIENT: If x is positive, then statement (2) says that 1 < xy (we do not flip the sign when cross multiplying). Thus, xy > 0.

Hi Bunuel ? Could you please explain how could we infer 1 < xy from statement 2 ??
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Joined: 22 Aug 2013
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Re: If x ≠ 0, is xy > 0?  [#permalink]

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santro789 wrote:
Bunuel wrote:
Bunuel wrote:
If x ≠ 0, is xy > 0?

(1) x > 0
(2) 1/x < y

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

(1) INSUFFICIENT: This tells us nothing about the sign of y.

In evaluating Statement (2), you might be tempted to assume that x must be positive. After all, we just read information in Statement (1) that tells us that x is positive. Besides, it is natural to assume that a given variable will have a positive value, because positive numbers are much more intuitive than negative numbers.

Instead, if we follow Principle #4, we will actively try to violate Statement (1), helping us expose the trick in this question.

(2) INSUFFICIENT: If we contradict Statement (1) to consider the possibility that x is negative, we would realize that it is necessary to flip the sign of the inequality when we cross multiply. That is, if x < 0, then 1/x < y means that 1 > xy, and the answer to the question is MAYBE.

(1) & (2) SUFFICIENT: If x is positive, then statement (2) says that 1 < xy (we do not flip the sign when cross multiplying). Thus, xy > 0.

Hi Bunuel ? Could you please explain how could we infer 1 < xy from statement 2 ??

Hi

We have NOT inferred 1 < xy ONLY from statement 2. This has been inferred from both statements combined, as explained in the official solution as quoted by you.
See first statement tells us that x > 0. So 1/x > 0. Now, as per second statement, y > 1/x. Since 1/x is positive, and y > 1/x, so y is also positive.
Both x and y are thus positive. So we can conclude that xy > 0
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Re: If x ≠ 0, is xy > 0?  [#permalink]

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santro789 wrote:
I dint get this - How combined statements are sufficient? Could someone explain by inserting values ?

Hi

I have already explained as a reply to you. But let me try using values. We know that x is not 0, and we need to know whether xy > 0 or not. Now product of xy will be positive when either both x/y are positive, or both x/y are negative.

(1) x > 0
But we dont know about y so we cannot say. Assume x=1, now if y =-1, xy = 1*-1 which is negative. but if y =2, xy = 1*2 which is positive.

(2) 1/x < y
If x is positive say 2, then 1/2 < y. Since y > 1/2 it has to be positive.
But if x is negative say -2, then -1/2 < y or y > -1/2. y is greater than -1/2, but it could lie between -1/2 and 0 (thus negative) or it could be 0, or it could be greater than 0 (thus positive). So we cant say whether product of xy will be positive or 0 or negative.

Combining the two statements, x>0. Take any positive value of x. If x=2, then 1/x = 1/2, y > 1/2 so both x and y are positive. Take x=1/500, then 1/x = 500, y > 500 so again both x and y are positive. So product x*y will always be positive. Re: If x ≠ 0, is xy > 0?   [#permalink] 16 Dec 2017, 11:02
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