fangtray wrote:is xy>0?

1) x-y > -2
2) x-2y < -6

i know i can brute force this problem and try all sorts of different numbers to find the right answer, but during the actual exam, you can make mistakes this way, and it takes a lot of time...could someone show me a good way of doing this problem and problems that are similiar??

First notice that the target question asks us whether or not the product of x and y is positive. To answer this, we want to know the signs (negative or positive) of the two variables.

We can rephrase the target question as, "Are both variables the same sign?"

Okay from here, we need to recognize right away that statements 1 and 2 cannot be sufficient.
We know this because we can't learn anything (sign-wise) about two variables if we're given information about their difference.

For example, if I tell you that x-y = 100, what can we say about the signs of x and y? Nothing. All we know is that x is 100 greater than y.
Similarly, if I tell you that x-y = -100, we can't conclude anything about the signs of x and y. All we know is that y is 100 greater than x.

So, at this point, we already know that the correct answer here must be C or E.

Let's examine the statements combined.
x-y > -2
x-2y < -6

Since both inequalities have an x, let's isolate x in both of them to get:
y-2 < x
x < 2y-6

Aside: Notice that I rewrote them so that the 2 inequality symbols are pointing in the same direction.

Now we can combine these inequalities to get: y-2 < x < 2y-6
Next, remove the x to get: y-2 < 2y-6
Then subtract y from both sides and add 6 to both sides to get: 4 < y
Great, we now know that y is positive.
Also, if y-2 < x (and y>4), then we know that x must also be positive

Since we now know that x and y are positive, we can answer the target question with certainty.

So, the answer is C

Cheers,
Brent