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.

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)
