1)x – y > – 2
Let x = 5, y = -2, xy = -10 < -2
Let x = 5, y = 2, xy = 10 > -2
Insufficient.

2) x - 2y < -6
Let x = 2, y = 6, xy = 12 > -2
Let x = -2, y = 6, xy = -12 <-2
Insufficient.

When combined, it gets a bit complicated. In typical 2 inequalities setups, you will need to add the two inequalities to eliminate one of the variables like what you do when solving 2 linear equations with 2 unknowns. You can only add to inequalities only if the inequalities are the same sign.

Here
x - y > -2 (1)

x - 2y < -6
-(x - 2y) > 6
-x + 2y > 6 (2)

Add inequalities (1) and (2) yields
y > 4

From (1)
x > y - 2
then x > 2

xy > 8 > -2

Answer is C.