1. Well, I believe we're better off trying a few numeric examples for this one. Say m = 5 and z = 1. This means that m - 3z = 2 > 0 and m + z = 6 > 0.
Now let's take another example: Say m = 5 and z = -6.
m - 3z = 5 + 18 = 23 > 0. But m + z = 5 - 6 = -1 <0. So 1 is not sufficient.

2. is not sufficient using the same line of thought as above.

Now, let's take the two statements together. Adding the two is equivalent to z > 0. Since z is positive, this means that, using statement 1, m can only be positive as well. Adding the two positive numbers will always be a positive number, so both statements taken together are sufficient.

Agree with C
Just draw the number line.

Stem 1 can guve you positive and negative meaning for m and z and their sum can be as positive as negative.

Stem 2: the same approach as with stem1.

Combined:
3z < m < 4z
Ths can be only if both are positive. So, sum is also positive.

4meonly wrote:Agree with C
Just draw the number line.

Stem 1 can guve you positive and negative meaning for m and z and their sum can be as positive as negative.

Stem 2: the same approach as with stem1.

Combined:
3z < m < 4z
Ths can be only if both are positive. So, sum is also positive.

Both explanations are master class and less time consuming. Remember, it's always worthwhile to use permissible number(s), in case you cannot visualize a rule, this is why I found the first work as awesome. The second work is accomplished by keeping rules in mind, which is always admirable on GMAT.