Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$. Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. 
I do not understand how to go about completing this problem or even where to start.

Without loss of generality, let $y=x+k$ for some nonnegative number $k$. Then,
$$
\frac{x+(x+k)+|x-(x+k)|}{2} = \frac{2x+2k}{2} = x+k = y
$$
which is equal to $\max(x,y)$ by the assumption.
