Question: assume $l$ is a particular integer. Is $2k-1$ odd?
$n = 2k+1$
$(2k-1) = 2k+1$
$2k = 2k+2$
$k = k+1$
or
$n = 2k+1$
$(2k-1) = 2k+1$
$2k - 2= 2k$
$k = k-1$
The answer I was given from my prof is "$k-2$" but I realized it could be $k+1$ or $k-1$, which one is it?
To prove an integer is odd, when you divide it by $2$, you get $1$ as the remainder, in particular, if a number, $n$ is odd, you can represent it using $n=2m+1$ for some integer $m$.
If you want to show that $2k-1=2m+1$ for some integer $m$ and you want to solve for such $m$. then we have $2k-2=2m$ and we conclude that $m=k-1$.
Remark: Note that I introduce notation $m$ to avoid reusing the notation $k$. Reusing symbols which come from different context is confusing and should be avoided. It is also unclear what do you mean by $k-2$ is an answer from your prof, the answer is simply, it is odd or it is even.