What will be max$\{0,a+b\}$?

Question

I know max$\{0,a+b\}=\mbox{max}\{0,a\}+\mbox{max}\{0,b\}$, whenever $a$ and $b$ share same sign.What happens if they have different sign? Can we have some identity?

Answer 1

$$\max\{\alpha ,\beta \}=\frac{\alpha +\beta +|\alpha -\beta |}{2}.$$