Use the definition of absolute value:
[tex]|x|=\begin{cases}x&\text{for }x\ge0\\-x&\text{for }x<0\end{cases}[/tex]
This means for [tex]-2<x<3[/tex], we have
- [tex]|x-3|=-(x-3)=3-x[/tex] because [tex]x<3[/tex]
- [tex]|x+2|=x+2[/tex] because [tex]x>-2[/tex]
- [tex]|x-5|=-(x-5)=5-x[/tex] because [tex]x<5[/tex]
So we end up with
[tex]y=(3-x)+(x+2)-(5-x)\implies y=x[/tex]
over the given interval.
