Prove relation is a function

Question

I want to prove this relation $$2y+|y|=3x$$ is a function. It is not difficult to prove by definition ,but I was searching for a beautiful idea to show that .I did like below . Is there more Idea to show this without using derivation ?

Any hint or Idea ? Thanks in advanced .

Answer 1

$$2y+|y|=3x \\\to \begin{cases}y \geq 0 & 2y+y=3x & y=x \to x\geq0 \\y <0 & 2y-y=3x & y=3x \to 3x<0 \to x<0\end{cases}$$ and plot two part ,show that this a function

Answer 2

We know that there is a unique $y$ for $x=0$.

If we take the derivative with respect to $x$ we find that

$$2y^\prime+\frac{y}{\vert y\vert}\cdot y^\prime=1 \text{ for }y\ne0$$

therefore

$$ \text{For }y\ne0,\quad y^\prime=\frac{1}{2+\frac{y}{\vert y\vert}}\ge\frac{1}{3}$$

Thus, it is monotonically increasing on $(-\infty,0)$ and $(0,\infty)$.