Bunuel
Is xy>0?
Note that question basically asks whether \(x\) and \(y\) have the same sign.
(1) x-y > -2 --> we can have an YES answer, if for example \(x\) and \(y\) are both positive (\(x=10\) and \(y=1\)) as well as a NO answer, if for example \(x\) is positive and \(y\) is negative (\(x=10\) and \(y=-10\)). Not sufficient.
(2) x-2y <-6 --> again it' easy to get an YES answer, if for example \(x\) and \(y\) are both positive (\(x=1\) and \(y=10\)) as well as a NO answer, if for example \(x\) is negative and \(y\) is positive (\(x=-1\) and \(y=10\)). Not sufficient.
You can get that the the two statement individually are not sufficient in another way too: we have (1) \(y<x+2\) and (2) \(y>\frac{x}{2}+3\). We are asked whether \(x\) and \(y\) have the same sign or whether the points (x,y) are in the I or III quadrant ONLY. But all (x,y) points below the line \(y=x+2\) (for 1) and all (x, y) points above the line \(y=\frac{x}{2}+3\) cannot lie only in I or III quadrant: points above or below some line (not parallel to axis) lie at least in 3 quadrants.
(1)+(2) Now, remember that we can subtract inequalities with the signs in opposite direction --> subtract (2) from (1): \(x-y-(x-2y)>-2-(-6)\) --> \(y>4\). As \(y>4\) and (from 1) \(x>y-2\) then \(x>2\) (because we can add inequalities when their signs are in the same direction, so: \(y+x>4+(y-2)\) --> \(x>2\)) --> we have that \(y>4\) and \(x>2\): both \(x\) and \(y\) are positive. Sufficient.
Answer: C.
Hey Bunuel!
I am a bit slow when it comes to thinking of numbers to solve DS Questions like these. I am quite strong in visualising the graphs when it comes to equations. So I started by plotting both the lines after quickly ruling out answer choices (A) and (B). After plotting, I selected the common area i.e the intersection of x - y > -2 and x - 2y < -6 and the common area of these two equations lies completely in the first quadrant, therefore, both x and y will always be positive and we can conclude that (C) is the correct answer. Is there any downside to this approach? I know all the graphs of the basic equations that I have seen on 750, or 700 level questions, what other graphs should I know about if I want to solve problems like these using the graph approach?