1. help!

Is xy>0?

1) x-y>-2

2) x-2y<-6

Choose among the 5 possibilities:
A) STATEMENT 1 ALONE IS SUFFICIENT
B) STATEMENT 2 ALONE IS SUFFICIENT
C) THE 2 STATEMENTS TOGETHER ARE SUFFICIENT
D) EACH STATEMENT ALONE IS SUFFICIENT
E) THE 2 STATEMENTS TOGETHER ARE NOT SUFFICIENT

The textbook I'm practicing on gives C) as the answer, but I cannot figure it out. Could you please explain the right solving sequence?
thank you

2. Originally Posted by simone

Is xy>0?

1) x-y>-2

2) x-2y<-6

Choose among the 5 possibilities:
A) STATEMENT 1 ALONE IS SUFFICIENT
B) STATEMENT 2 ALONE IS SUFFICIENT
C) THE 2 STATEMENTS TOGETHER ARE SUFFICIENT
D) EACH STATEMENT ALONE IS SUFFICIENT
E) THE 2 STATEMENTS TOGETHER ARE NOT SUFFICIENT

The textbook I'm practicing on gives C) as the answer, but I cannot figure it out. Could you please explain the right solving sequence?
thank you
You have the condition $\displaystyle xy > 0$

So consider:
$\displaystyle x - y > -2$

$\displaystyle -y > -x - 2$

$\displaystyle y < x + 2$

So if x = 1 then y could certainly be -10. Thus we can't say $\displaystyle xy > 0$.

Let's try the second one:
$\displaystyle x - 2y < -6$

$\displaystyle -2y < -x - 6$

$\displaystyle y > \frac{x}{2} + 3$

Here, even if y is positive, x could well be negative. (Try x = -1, y = 10).

So neither condition by itself will guarentee that $\displaystyle xy > 0$

What about together? The easiest way to see this is to graph it. (See below.) Where the shaded regions overlap is the solution to the combined inequalities and is a region where both x and y are positive. Thus we have a requirement that $\displaystyle xy > 0$, so both conditions are required and sufficient.

-Dan

3. a big big big thank you , I'm sorry for double posting.