If $x+|x|+y=5$ and $x+|y|-y=6$, what is the value of the sum $x+y$?
(a) -1
(b) 11
(c) $\frac{9}{5}$
(d) 1
(e) -11
The correct option is (c).

1st Case: If $x \leq 0$, then $|x|=-x$ and, by the first equation, we have $x+(-x)+y=5$, that is, $y=5$. Substituting this value into the second equation, we get $x=6$, which is not possible since we are assuming $x \leq 0$. Therefore, there is no solution in this case $x \leq 0$.

$2^{\text{nd}}$ Case: If $y \geq 0$, then $|y|=y$ and, by the second equation, we have

$$
x+y-y=6
$$

that is, $x=6$. Substituting this value into the first equation, we get $y=-7$, which is not possible since we are assuming $y \geq 0$.

3rd Case: If $x>0$ and $y<0$, then $|x|=x$ and $|y|=-y$. By the first equation, we have $2 x+y=5$ and, by the second, $x-2 y=6$. Multiplying $2 x+y=5$ by 2 and adding to $x-2 y=6$, we get $5 x=16$, so $x=16 / 5$ and it follows that $y=5-2 x=-7 / 5$. Thus, $x+y=9 / 5$.