It would be helpful to know what the rest of the question is as well as the answer choices. Is this a data sufficiency?

As a statement, this is only true if X is positive. If X is negative, then that statement is false.

hey can you kindly explain in steps ,how you arrived at the answer..

apoorvasrivastva, your approach is good for case 1, however for case 2 it will be -1 < x < 0 as nplaneta posted earlier.

The reason being...
(x^2 - 1) / x > 0
When x < 0 then:
(x^2 - 1) < 0 [multiplying both sides by x (which is a -ve number, hence flipping the inequality)]
x^2 < 1
now as we are considering x < 0 then, when we take square root of both side:
sqrt(x^2) < sqrt(1)
x < 1 [but x is a negative number so we need to multiply both sides by -1]
-x > -1
x > -1 [left side became positive due to negative x and -1]
and since the case is for x < 0 , therefore, the solution is -1 < x < 0

The solution for x > 1/x being:
x>1 (when x>0) and -1 < x < 0 (when x<0)

For reference and review of inequality principles, see basic-mathematical-principles post by HongHu in this forum [sorry I can't post the direct link, as the forum isn't allowing me cos I am new member]
This exact problem is explained in this post as well.

~Cheers
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[EDIT - REMOVED BY THE USER]

Right, our equation was (x^2-1)/x > 0, so to get rid of x from the denominator you have to multiply both sides of the inequality by x, that is (x^2-1)*x/x > 0*x => (x^2-1)/x < 0 and the sign is flipped because x < 0. Multiplying both sides by x is what you do when we simply "cross-multiply" but we can only safely cross-multiply when it is equality but not in inequality.

Secondly, upon solving our equation we arrived at:
x < 1

Now x in this equation is -ve, since we are considering a case where x < 0. So if we leave it like x < 1 then the x in this case is not comparable with x in our solution for the case where x > 0. So in order to compare apples with apples, we need to get x to mean the same thing for both cases.
Therefore, we multiply both sides by -1 (which flips the sign again) to arrive at:
x > -1

Also, try picking few numbers in and outside the boundaries in the original equation, which may also help.


On a separate note, I think if you have doubts on questions in a particular post then its better to post your doubts on that post itself, so as to keep all responses on the issue together, instead of starting the new post. I don't know if this is the policy on this forum or not, may be moderator can clarify?
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The three most significant times in your life are:
1. When you fall in love
2. The birth of your first child
3. When you prepare for your GMAT

Yes the faster the better, thanks for your approach...

@Spidy001, I don't think you can multiply by x with the inequality sign intact without knowing the sign of x.

See this solution :

=> x - 1/x > 0

=> (x^2 -1)/x > 0

So the sign of numerator and denominator are same and if x > 0 then x^2 - 1 > 0 and if x < 0 then x^2 - 1 < 0

(x+1)(x-1) > 0 and (x - 1) (x+1) < 0

so x > 1 Or x < -1 AND -1 < x < 1

Hnce x > 1 OR -1 < x < 0
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