Points on a plane satisfying an equation

[brotherbobby]

Homework Statement

Indicate the points on the place $xOy$ which satisfy the equation : $\mathbf{y+|y|-x-|x| = 0}$.

Relevant Equations

The modulus of a variable $|x| = x\; \text{if}\; x\geq 0$ and $-x\; \text{if}\; x\leq 0$

We can write the equation given as $y+|y| = x+|x|$

This has a few conditions.

(1) If $\underline{y\geq 0\; \text{and}\; x\geq 0}$, we obtain $2y = 2x \Rightarrow \boxed{y = x}$.
(2) If $\underline{y\geq 0\; \text{and}\; x < 0}$, we obtain $2y = 0 \Rightarrow \boxed{y = 0}$.
(3) If $\underline{y < 0\; \text{and}\; x\geq 0}$, we obtain $2x = 0\Rightarrow \boxed{x = 0}$
(4) If $\underline{y < 0\; \text{and}\; x < 0}$, we obtain the trivial solution $0 = 0$. But since 0 =0 always, this means all values of $y<0\;\text{and}\; x<0$ are solutions to the equation.

I plot the graph alongside for the solution and regions.

On the left is the solution from the book.

Though my answer looks correct, is the reasoning alright?

 

[FactChecker]

That looks good to me.

 

[haruspex]

You can slightly simplify the logic by considering just two cases, x+|x| zero or nonzero.