How to correctly factor out a $\pm$ from a number that has a $\mp$ in front?

Question

A very simple example is the following; suppose I wish to factor out $\pm$ from $\mp 7$, then by my logic $$\mp 7= \color{blue}{\pm \mp} (\mp 7)=\begin{cases}+(-(-7)) & =+7\\ -(+(+7)) & =-7\end{cases}=\pm 7\ne \mp 7\tag{*}$$

So something has gone wrong, as equality is not satisfied in $(*)$. The reason why I wrote $\color{blue}{\pm \mp}$ is because if I factor out a $\pm$ then to compensate I must include a $\mp$ sign (at least I thought).

Just like if I took a factor of $x$ out of $$1+x=x\left(\frac{1}{x}+1\right)$$ so that $$x\times \frac{1}{x}=1$$

But, this same logic does not seem to apply to $(*)$.

Clearly, I am missing something very simple, but right now I can't understand what I'm doing wrong. Could someone please explain how to factor out $\pm$ while maintaining equality?

Answer 1

Note that

$$\color{blue}{\pm1\cdot \mp1} =-1 \implies \color{blue}{\pm \mp} (\pm 7)=\mp7$$

Answer 2

In a ring you have the following rules: $a\cdot 0 = 0 = 0\cdot a$, $-(-a)=a$, $(-a)b = a(-b)=-(ab)$, and $(-a)(-b)=ab$, where $-a$ is the additive inverse of $a$. From here, everything clarifies.