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Notice that the question basically asks whether x and y have the opposite signs.

(1) \(\frac{1}{x}<\frac{1}{y}\) --> \(x\) and \(y\) can have the same sign (\(x=2\) and \(y=1\)) as well as opposite signs (\(x=-2\) and \(y=1\)). Not sufficient.

(2) \(x>0\) --> no info about \(y\). Not sufficient.

(1)+(2) As from (2) \(x>0\) then \(\frac{1}{x}<\frac{1}{y}\) to hold true, \(y\) must also be positive (left hand side, \(\frac{1}{x}\) is positive and right hand side, \(\frac{1}{y}\), to be greater than that, must also be positive). So \(x\) and \(y\) have the same sign, they are both positive: \(xy>0\). Therefore the answer to the question is NO. Sufficient.

Hey Bunuel, Can you please let me know where I'm going wrong in the following method: Statement1: 1/x < 1/y Subtract 1/y from both sides: we get (y-x)/xy <0. Therefore, either xy<0 or (y-x)<0. If xy<0 then the answer to our question is YES both are opposite signs. If y-x<0 -> y<x, then both can have same sign or opposite signs (ie.., y<x<0 OR y<0<x OR 0<y<x) Therefore, Statement1 is INSUFFICIENT. Statement2: 0<x so x is Positive but no information about y, So statement 2 is INSUFFICIENT. (1) and (2) together: We still have y<0<x or 0<y<x Therefore, Both statements together are INSUFFICIENT. So E. Thanks for your help!

Hey Bunuel, Can you please let me know where I'm going wrong in the following method: Statement1: 1/x < 1/y Subtract 1/y from both sides: we get (y-x)/xy <0. Therefore, either xy<0 or (y-x)<0. If xy<0 then the answer to our question is YES both are opposite signs. If y-x<0 -> y<x, then both can have same sign or opposite signs (ie.., y<x<0 OR y<0<x OR 0<y<x) Therefore, Statement1 is INSUFFICIENT. Statement2: 0<x so x is Positive but no information about y, So statement 2 is INSUFFICIENT. (1) and (2) together: We still have y<0<x or 0<y<x Therefore, Both statements together are INSUFFICIENT. So E. Thanks for your help!

\(\frac{y-x}{xy} <0\) means that y-x and xy have the opposite sign: +- or -+.

When combined we know that x is positive. Now, if y were negative, then \(xy<0\), thus \(y-x\) must be positive, but in this case \(y-x=negative-positive=negative\), thus this case is not possible, y is NOT negative --> y is positive --> \(xy=positive\).

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
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Is xy < 0 ? It's sometimes helpful to think of disproving rather than proving the statement. For xy to be negative, x and y have to be opposite signs. (1) 1/x < 1/y For this statement consider x and y being both positive or both negative. For example, (1/4) <(1/2) or (-1/2) < (-1/4). Not sufficient.

(2) x > 0 We're given nothing about y so it could be positive as well. Not sufficient.

Multiply both sides of (1/x) < (1/y) by x to give (x/y) > 1 From statement 2 we know that x is positive. For x/y to be greater than one y also has to be positive. Sufficient (no) C _________________

statement 1=> not sufficient statement 2 => not sufficient combing them we get x/y>1 hence x/y>0 9as it is greater than 1 it must be greater than 0 ) so xy>0 (xy and x/y have the same sign)

Here to add to the explanation => X IS ALWAYS POSITIVE AND Y IS ALWAYS POSITIVE TOO => XY>0 the key to solving these question is to look out for a sufficient YES or a sufficient NO
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