How would you go about graphing a set of points (x,y) that satisfies the following equation? Could you explain how this is different from graphing any other equation? 
What confuses me is that there are two y-values and two x-values. 
The equation is: y+|y| = x+|x|

Split it into four equations:
$y,x \geq 0 \implies 2y =2x \implies y=x$
$x \leq 0 \leq y \implies 2y=0 \implies y=0$.
$y \leq 0 \leq x \implies 2x=0 \implies x=0$.
$y,x \leq 0 \implies 0=0$, so here this is true for all values of $x$ and $y$.
Hence, we take the graphs of these four functions and combine them to get the solution set:
$$
S = \{ y \leq 0 ,x \leq 0\} \cup \{ y =0\} \cup \{x=0\} \cup \{x=y\}
$$
The second and third set absorb into the first, so the answer would be:
$$
S = \{ y \leq 0 ,x \leq 0\} \cup \{x=y\}
$$
Hence $|x| + x = |y| + y$ if and only if $(x,y) \in S$.
