Determine whether the region $\text{Im}(\frac{z+1}{z-1}) \leq 3$ is open, closed or neither. Determine whether the region $\text{Im}\left(\frac{z+1}{z-1}\right) \leq 3$ is open, closed or neither.

My attempt and the memo for the question is below.
My answer is somewhat similar to the memo's except that I didn't have the $z \neq 1+0i$ condition.
Is this additional $z \neq 1+0i$ part correct?
While I was working on the solution it didn't seem immediately evident to me, that that condition was necessary.
If the memo is correct, do you have any tips on how I can spot such conditions in similar questions in the future?
Thank you.
Memo answer
My answer
See what happens when you substitute $z= 1+ 0i = 1\;$ into your inequality.
Then $$\text{Im}\left(\frac{z+1}{z-1}\right) = \text{Im}\left(\frac 20\right).$$
You can't do that, because $\frac 20$ is undefined; so we must forbid $z = 1 + 0i = 1$.
See what happens when you substitute $z= 1+ 0i = 1\;$ into your inequality.
Then $$\text{Im}\left(\frac{z+1}{z-1}\right) = \text{Im}\left(\frac 20\right).$$
You can't do that, because $\frac 20$ is undefined; so we must forbid $z = 1 + 0i = 1$.