Notice that if a > b, and c > d, then a+c must be greater than b+d. We can use that here, but we need to rewrite one of the inequalities first, so that both are in the same direction (>). We can multiply the second inequality by -1 (reversing the inequality)

x - y > -2
-x + 2y > 6

adding these:

y > 4

And since x - y > -2, x > y - 2, so x > 2. Since x and y are both positive, xy must be positive.

Alternatively, you can do 2-variable inequality problems graphically. Look at the first inequality:

x - y > -2, or rearranging:
y < x + 2

This inequality represents a region in the co-ordinate plane. The line

y = x + 2

is the boundary between the regions
y < x + 2 and
y > x + 2

You can draw that line, work out that when y < x + 2, the points must be below the line, and see all of the possible values for (x, y). We can do the same for the second equation, and thereby find the region where both inequalities must be true (they will be true for all points below the line y = x +2 which are above the line y = 0.5x + 3, and these points are only in the first quadrant). It's easier to see if you draw the picture.

Vignesh.4384 wrote:Thanks Ian.

Very useful ideas.

How would u rate this problem ? easy or tough ?

Seemed a bit diff at first but wasnt the case after seeing ur answer.

Well, I think most GMAT math questions seem easy once you see how to do them- it's seeing how to do them that can be tricky. I definitely don't think this question is an 'easy' GMAT problem - it's a two-variable inequality problem, which already makes it a bit complicated, and I don't think it's easy to arrive at the right answer using shortcut methods like choosing numbers. So my quick assessment- it's probably a medium-difficult problem. That said, it can be quite surprising to see which questions are actually calibrated as difficult questions on the test, and which ones as easy questions- it's sometimes surprisingly hard to judge.