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.
 A: You can try the two cases separately.
Assume $v$ is negative and then divide through by $v$ giving
$$ m > p > 0\;\;\;  (*)$$
This is contradicted by both statements (1) and (2). Hence by contradiction if (1) or (2) are true, then $v$ cannot be negative.
On the other hand if you assume $v$ is positive and then divide by $v$ you get
$$ m < p < 0\;\;\; (\dagger) $$ 
This is consistent with both statements (1) and (2). Hence we conclude that either  (1) or (2) or both will show that $v>0$. So I agree with your answer D they are both independently sufficient.
Finally if we have no information then we can't say whether (*) or $(\dagger)$ holds, so we do need at least statement (1) or (2) to be true. (and also note that $v$ can't be zero)
A: So when GMAT textbooks right this:
"One very important implication of this rule is: You cannot divide by an unknown (i.e., a variable) unless you are sure of its sign since you do not know whether you must flip the sign of the inequality. There are plenty of instances where you will know the sign of a variable and as a result, you can multiply or divide and know for sure whether you must flip the inequality sign. However, you must always ask yourself whether you know for sure the sign of the variable before dividing or multiplying when dealing with an inequality.
If 2x5y < 10y, what is the range of potential values for x?
You cannot divide by y or 5y since you do not know whether y is negative or positive and, as such, you do not know whether to flip the inequality."
Is this not true? You can divide by a variable but you have to text both scenarios?
A: 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
