In which quadrant of the coordinate plane does the point

I know this is late but I had to give my crack on it and, of course, so other people might be able to learn from this very great DS question.

Anyway,

The rephrase of the question asks us for the signs of x and y. If we know the signs of x and y then we'll be able to determine which quadrant does (x,y) lie.

On to the statements:
Statement (1) |xy| + x|y| + |x|y + xy > 0

There are two ways of attacking this:
1. By algebra (conceptual method)
We can test four cases:

(A) x < 0 and y < 0:
xy + (-x)(y) + xy + (-x)(y) > 0
xy - xy + xy - xy > 0
0 = 0
So this doesn't satisfy the equation

(B) x > 0 and y < 0:
xy + xy + (x)(-y) + (x)(-y) > 0
xy + xy - xy - xy > 0
0 = 0
So this doesn't satisfy the equation

(C) x < 0 and y > 0:
xy + (-x)(y) + xy + (-x)(y) > 0
xy - xy + xy - xy > 0
0 = 0
So this doesn't satisfy the equation

(D) x > 0 and y > 0:
xy + xy + xy + xy > 0
4xy > 0
This satisfies the equation. We conclude that x > 0 and y > 0 or x = + and y = +, which means that point (x,y) lies in quadrant I.
Statement (1) is sufficient.

2. By plugging in numbers:
x = + 3
x = - 3
y = + 3
y = -3

(A) x = -3 and y = -3

|(-3)(-3)| + (-3)|-3| + |-3|(-3) + (-3)(-3) > 0
9 - 9 - 9 + 9 > 0
0 = 0
Doesn't satisfy the equation.

This is the same as the above method. This helps if someone is having a hard time dealing with the "theoretical" side of modulus.

Statement (2) -x < -y < |y|

We break down the equation into two
(A) -y < |y| -> for this to be true, y should be +
(B) -x < -y multiplying both sides by negative 1 we have: x > y (since we know that y is positive, for this to be true x should be true)
Therefore, we know that x and y are positive. Statement (2) is sufficient.

Hope other people could chime in on this, especially statement (2). I am not convinced with myself. HAHA!

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-which-quadrant-of-the-coordinate-plane-does-the-point-86767.html