Solve y=|x|+m/x+4

$$y = | x | + \frac { m } { x } + 4$$

Solve for m

$$m=-x\left(|x|-y+4\right)$$

$$x\neq 0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{y^{2}-8y-4m+16}+y-4}{2}\text{, }&\left(y\geq 2\sqrt{m}+4\text{ and }y>4\text{ and }m\leq \frac{\left(y-4\right)^{2}}{4}\text{ and }m>0\right)\text{ or }\left(y=2\sqrt{m}+4\text{ and }m>0\right)\\x=\frac{\sqrt{y^{2}-8y-4m+16}+y-4}{2}\text{, }&m<0\text{ or }\left(y>4\text{ and }y\geq 2\sqrt{m}+4\text{ and }m\leq \frac{\left(y-4\right)^{2}}{4}\text{ and }m\geq 0\right)\text{ or }\left(y>4\text{ and }m\leq 0\right)\\x=\frac{\sqrt{y^{2}-8y+4m+16}-y+4}{2}\text{, }&\left(y\geq 2\sqrt{-m}+4\text{ and }y>4\text{ and }m\geq -\frac{\left(y-4\right)^{2}}{4}\text{ and }m<0\right)\text{ or }\left(y=2\sqrt{-m}+4\text{ and }m<0\right)\\x=\frac{-\sqrt{y^{2}-8y+4m+16}-y+4}{2}\text{, }&m>0\text{ or }\left(y>4\text{ and }y\geq 2\sqrt{-m}+4\text{ and }m\geq -\frac{\left(y-4\right)^{2}}{4}\text{ and }m\leq 0\right)\text{ or }\left(y>4\text{ and }m\geq 0\right)\end{matrix}\right.$$