My reasoning would be that since $x$ is negative, $x$ can be rewritten as $-1 \cdot y$ where $y$ is equal to $|x|$. Therefore we can write $a - x$ as $a - (-1) \cdot y = a + y$, which is not equal to $a + x$.
I'm asking because it seemed very natural to me to just turn the sign around, because we do that in school where for example $2 - (-4)$ is equal to $2 + 4$, but $4$ isn't equal to $-4$.
And if what I said is true, does it mean that I have to take this exact step while writing proofs as well, or is there a more elegant way to do it?