Let me try to stitch together what happens when we look at the two statements together. From S1, we got that \(m>3z\). From S2, we got that \(m<4z\), right?

Thus, we can combine those two facts to say \(3z<m<4z\). From there, extract that \(3z<4z\).

Option 1) From here, simply subtract \(3z\) from both sides; we explicitly see that \(z>0\).

Option 2) This is where the magic happens. Try to divide both sides by \(z\). Whoa, panic! You don't divide both sides of an inequality by a variable; are you crazy??! ...except that we end up with \(3<4\), which is, uh, true. Therefore, it was ok to divide by z, which tells us that z is positive.

If you have a copy of it, OG12 DS#11 illustrates a similar problem, using Option 1.

Now, at this point. Recall from S1 that \(m>3z\), and since \(z\) is positive, clearly \(3z\) is positive, thus \(m\) is positive. Finally, since \(m\) and \(z\) are both positive, clearly their sum is positive. We now have sufficiency.

I noticed an error in someone's explanation earlier; they concluded that if \(m>3z\), then it must also be true that \(m>z\). This would only hold if we were guaranteed that \(z\) were positive, so that we can chain that \(m>3z>z\) and then get \(m>z\). However, if \(z\) is negative, then we can't be sure. For example, let \(m=-2\) and \(z=-1\). Thus, \(-2 = m > 3z = -3\), but it's not true that \(-2 = m > z = -1\).

Furthermore, to be perfectly honest, it took me a bit more to prove to myself that either statements 1 or 2 were insufficient! I personally do not declare INS until I either realize that some information is OBVIOUSLY missing (OG12 DS#15 S1, for example), or until I see proof through examples that the QUESTION can be answered multiple ways.

For example, in S1, to confirm insufficiency, I said let \(m=10\), and \(z=3\). This satisfies the given condition (\(10-3*3>0\)), and then the answer to the question that is asked is YES (\(10+3>0\)).

Then, I tried another one; I said what if z is negative? Ok, let's try \(m=10\), and \(z=-10\). The condition still holds (\(10+3*(-10)>0\)), but now the answer to the question is NO (\(10+(-10)>0\)).

Now, I safely declare insufficient. Similarly, the test cases \((m,z)=(1,1)\) and \((m,z)=(-1000,1)\) establish that statement 2 is insufficient.

As someone who has an extremely rigorous formal math background, I don't move forward until I get a

positive signal that something is insufficient. I know that "insufficient" and "insufficient for me!" don't mean the same thing

I always have the question in mind "what does insufficiency look like?"

Here's an example I just made up:

**Quote:**

If x and y are integers, what is the ratio of x to y?

1) 3x + 5y = 20

The easiest way to prove that this is insufficient is just to say, well, what if \((x,y)\) are \((5,1)\)? \(15+5=20\), check, then the ratio is \(5/1=5\). But what if \((x,y)\) are \((0,4)\)? \(0+20=20\), check, then the ratio is \(0/4=0\). Different. Insufficient. You can't argue that.