For $x, y \in \mathbb{R}$, prove that $\max(x, y) = \frac{x + y + |x - y|}{2},$ and $\min(x, y) = \frac{x + y - |x - y|}{2}$. Prove that for all real numbers $x$ and $y$, 
$$\max(x, y) = \dfrac{x + y + |x - y|}{2},$$
and 
$$\min(x, y) = \dfrac{x + y - |x - y|}{2}.$$
For any real number $x$, the absolute value of $x$, denoted $|x|$ is defined as follows:
\begin{equation}
|x| = \begin{cases} x; & \text{ if } x \geq 0 \\
-x; & \text{ if } x< 0
\end{cases}
\end{equation}
What I understand from this is that $|x| = x$ if $x \geq 0$ or $|x| = −x$ if $x<0$. Other than that I don't really know how to start this. 

\begin{align}
   \max\{x,y\} + \min\{x,y\} &= x + y \\
   \max\{x,y\} - \min\{x,y\} &=  |x-y| \\ 
\text{Adding we get} \\
2\max\{x,y\} &= x + y + |x-y| \\
\text{Subtracting we get} \\
2\min\{x,y\} &= x + y - |x-y| \\
\end{align}
