Your confusion lies in thinking that Addition supersedes Subtraction. The actual precedence set by "PEMDAS" is
- Parenthetical terms first
- Any multiplication OR division (equal precedence!)
- Any addition OR subtraction (equal precedence!)
This is why I suggested you think of it as PE(MD)(AS).
This is nothing more than a convenient agreement and not some mysterious law of nature. It allows us to write $3x-y$ without ambiguity, as it means $(3x)-y$ and not $3(x-y)$. If we mean the latter, we have to say it.
So, for example,
5\cdot 3 -2= 15-2=13
and NOT $5 \cdot 3-2=5$. If I wanted the latter I have to bracket it off to make it respect PEMDAS: $5\cdot (3-2) = 5 \cdot 1 =5$.
In general, when a "tie" in precedence comes, we operate left to right. For your case, $9-4+1=5+1=6$ since the subtraction is done first. You gave the second (addition) precedence, when really the subtraction gets done first, since they have equal precedence and the left-most one comes first.
To be honest, even this is sort of rigid. I can even do your addition first, as long as I respect the minus sign the right way. For instance
9 - 4 + 1 = 9 + (-4+1) = 9 + (-3) = 9-3 =6.
For some other examples,
2\cdot 5 - 7 \cdot 3 + 4 = 10 - 21 + 4 = -11 + 4 = -7.