<s>[INST] 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. 
 [/INST] 
If $\displaystyle x>y$ then $x-y>0$:

$|x-y|=x-y$
$\max(x,y) = (x+y+x−y)/2 = x$
$\min(x,y) =  (x+y-(x−y))/2 = y$

If $\displaystyle x<y$ then $x-y<0$:

$|x-y|=y-x$
$\max(x,y) = (x+y+y-x)/2 = y$
$\min(x,y) =  (x+y-(y-x))/2 = x$

If $x=y$ then:

$|x-y|=0$
$\max(x,y) = (x+y)/2 = x=y$
$\min(x,y) =  (x+y)/2 =x=y$
</s>