is xy > 0?
1) x - y > -2
2) x - 2y < -6
Question basically asks whether x and y have the same sign.
(1) y<x+2, can we find the pair of x and y with different signs satisfying the inequality given? Sure x=1 y any negative value. Can we find the pair of x and y with the same sign satisfying the inequality given? Sure x=5 y=1. So not sufficient to conclude whether x and y have the same sign or not.
(2) y>x/2+6. The same here. We can find x and y with different signs, as well as the same sign to satisfy inequality given. Not sufficient.
(1)+(2) Remember we can subtract inequalities with the signs in different direction. (1)-(2) 0<x/2-4 --> 8<x. So we get that x is positive and we know from (2) that y>x/2+6. x/2+6 is some positive value and y is more than that positive value, or in another words y is positive too. Sufficient.
The problem above can be solved in different way:
(1) y<x+2 represent all x,y points which are below the line y=x+2. If you draw this line you'll see that this are consists of the points which are in all quadrants. Or there are xy points with all combinations of x and y: positive x negative y, positive x positive y etc. and we are looking whether x and y are eother from the I quadrant or from the III. Not sufficient.
(2) y>x/2+6. Again if you draw the line y=x/2+6, the points satisfying the inequality given would be above of this line. The points would be in quadrants I, II, or III. Not sufficient.
(1)+(2) The area you'll receive from the two inequalities y<x+2 and y>x/2+6 (below the first and above the second line) will be consisting from the points which are only I quadrant, which means that x and y are both positive. Hence xy is positive. Sufficient.
as a review..subtract with signs in different directions and add with signs in same direction?