Is x < y ?

(1) x^2 - y^2 < 0
(2) -x - y < 0

St1: x^2 - y^2 < 0 --> |x| < |y|. Not sufficient as we do not know the signs of x and y.

St2: x + y > 0
x and y are positive --> x > y or y > x
or
either x or y is positive --> The positive value must be greater than the other value.
Not Sufficient.

Combining St1 and St2, we know that |y| > |x| and x + y > 0.
If |y| > |x| and x and y are positive, then y > x
If |y| > |x| and one value is positive, then the value, which has a greater number, must be positive --> y > x.
In both the cases we get a definite yes answer for the question is x < y.
Sufficient.

Answer: C

x < y?

stmt-1:
x^2 - y^2 < 0
|x| < |y|
here x or y or both can be postitive or negative. so we cannot decide if x < y or not.

stmt-2:
-x-y < 0
-x < y

say x=2, y=1 then -2 < 1, so x>y. so answer to the given question is NO.
say x=-1, y=2 then 1 < 2, but here x < y. so answer to the given question is Yes.

stmt-1 + stmt-2:
|x| < |y| AND -x < y then is x < y?
x=1, y=2 satisfies both |x| < |y| AND -x < y. in this case x < y (1<2). so answer is YES.
x=-1,y=2 satisfies both |x| < |y| AND -x < y. in this case x < y (-1<2). so answer is YES.

Sufficient!
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x^2 - y^2 < 0;

x^2 < y^2;

|x| < |y|.

This means that y is further from 0 than x is. Which is not enough to get whether x < y. Basically we can have the following cases:

----y----x----0--------------
----y---------0----x---------
---------x----0---------y----
--------------0----x----y----
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Hi ScottTargetTestPrep and Bunuel, thanks for your explanations.

What values could we plug into statement 2 to make it agree with the constraints?

I didn't think to multiply by -1, making it easier to eliminate statement 2, so I'm trying to see what I missed in my plug-in approach
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Damn. I realise how silly this mistake was. I didn't write "-" in front of the x i.e. -x-y so was wondering how the hell you could get x-y<0
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@ScottTargetTestPrep@Bunuel

Please evaluate the attached problem, the work done on this same question. I tried not to do it by putting values.

My pleasure.
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Official explanation

Explanation: (1) The values 2 and 3 will give us a YES answer to the question stem, but the values 2 and -3 will give us a response of NO. Another way to approach this is to realize that you can factor the left hand side of the inequality to get (x - y)(x + y) < 0. This means that either the first term, (x - y), is negative or the second term, (x + y), is negative, but not both. If the first term is negative, then YES is answer to the question. However, if the second term is negative, the answer could be YES or NO. Accordingly, this statement is insufficient.

(2) You can simplify this by multiplying both sides by -1, which means you need to flip the direction of the inequality. You'll now have x + y > 0. This, however, doesn't tell us which value is larger, and this statement is insufficient.

Together, we know that (x - y)(x + y) < 0 and x + y > 0. This means that x - y < 0 . Remember, for the product of two values to be negative, one of them and only one of them must be negative. Because we know that (x + y) must be positive, then (x - y) < 0. Simply adding to each side of that inequality, we find that x < y, and know that the statements together are sufficient. Accordingly, the answer is C.