If $mv < pv < 0$, is $v > 0?$ (1) $m < p$ (2) $m < 0$ Is there a way to simplify this equation and not rely on testing numbers via trial and error?
If $mv < pv < 0, is v > 0$?
(1) $m < p$
(2) $m < 0$
We have to figure out if statement 1 by itself is sufficient to answer this question, or if statement 2 is sufficient by itself to answer this question, or if both statements combined are necessary to answer this question, or if both statements independently are needed to answer this question, or if neither statement is sufficient.
So, I know we can't divide by v since we do not know if v is negative or positive. but can we divide by v and evaluate two cases?
Can we simply the stem of the question to:
m < p < 0 and
m > p > 0?
My answer is D, they are both independently sufficient.
For statement 1, if m < p and they are positive numbers, then there is no way for the inequality to work. Say m = 2 and p = 3, there is no value of v that works... so we can't test this case right? However, if m = -3 and p = -2, then V = 1 and V is > 0. Sufficient.
statement 2: I used the same numbers. Sufficient. I can't find a negative number for V that makes inequality work.

yes, on the gmat, you can divide by a variable, but you always have to keep in mind that you will need to test both scenarios.  This adds extra complications to the problem solving, which is probably why the book told you not to do it.  In general, I would avoid it if possible.  For this scenario, however its kind of necessary, because it shows you that:
if v>0, then if we divide by v, mp>0
(1) tells us that m0, therefore this one is sufficient
(2) tells us that m<0, that also means that v must be >0, because if v<0 then vm would be > 0, because to negatives makes a positive.  Therefore, this one is also sufficient and the answer is d
