Question

Rewrite without absolute value for the given conditions: y=|x−3|+|x+2|−|x−5|, if −2 < x < 3

Answer 1

Use the definition of absolute value:

`|x|=\begin{cases}x&\text{for }x\ge0\\-x&\text{for }x<0\end{cases}`

This means for
`-2<x<3`, we have

• 
  `|x-3|=-(x-3)=3-x` because
  `x<3`

• 
  `|x+2|=x+2` because
  `x>-2`

• 
  `|x-5|=-(x-5)=5-x` because
  `x<5`

So we end up with

`y=(3-x)+(x+2)-(5-x)\implies y=x`

over the given interval.