I know that there are some discussion over the definition over absolute value in stack-exchange. However, I did not satisfied. It is generally given that $$ \left| y \right| = \begin{cases} \phantom{-}y, & y \geq 0 \\ -y, & y < 0 \end{cases} $$
However, I think that this definition is a little bit dangerous, because it may give rise to wrong answer in some questions such that
Write all possible values of $x$ in $|2x-6|=6-2x$
If students use the foregoing definition, they will write $-1,-2,-3,\ldots$. However, $0$ should have been included.
I know that this will seem ridiculous to most of you, but when we imagine ourselves like a primary or secondary school students, it may cause problem.
So, my question is why most of book use this definition instead of clearer one such that $$\left| y \right| = \begin{cases} \phantom{-}y, & y \geq 0 \\ -y, & y \leq 0 \end{cases} $$
I fail to see what's non-clear about the usual definition. The definition that you suggest is logically correct, but may put it the mind of the students that there is no problem in defining a function with an expression such as$$f(x)=\begin{cases}\text{some expression}&\text{ if }x\geqslant a\\\text{some other expression}&\text{ if }x\leqslant a.\end{cases}$$And there is no problem in the definition that you have suggested. But, in general, you can't do that.