duongthang wrote:Pls, tell me how to solve quick

is xy>0?
1, x-y>2
2, x-2y<-6

This question is testing your knowledge of how positive and negative numbers behave.
First recognize that if xy>0 then x and y must be either both positive or both negative.
So, the question can be rewritten as "Do x and y have the same sign?" (both pos or both neg).

1) We can rewrite this as y+2 < x
Aside: I always rewrite my inequalities so that the greater value is on the right. Then the values are listed in ascendiing order.
Notice that, since we also know that y < y+2, we can add this to the inequality above to get: y < y+2 < x
From here we can see that x is greater than y.
This is not enough information to answer our question.

2) We can rewrite this as x < 2y-6
We can see that this isn't enough to answer our question (y could be positive and x could be negative, or they both could be positive)

(1&2)
We can combine both inequalities to get: y < y+2 < x < 2y-6
From this, we can conclude that y < 2y-6
From here we know that y must be positive. If y were negative, it would be impossible for 2y-6 to be greater than y.
Proof: Assume that y is negative and notice that this leads us to a contradiction. First, if y is negative, then 2y<y. And since 2y-6 < 2y, we would have to conclude that 2y-6 < 2y < y. In other words, it would have to be true that 2y-6 < y.
Our inequality y < y+2 < x < 2y-6 tells us the opposite is true.
So, y must be positive.
From statement (1), we know that y<x, so x must also be positive.
If x and y are both positive, we can answer our question.