Question

Solve `|x-1|-|2+x|=3` ?

Answer 1

`x = -5`

Explanation:

`|x-1|-|2+x|=3`

Making `y = abs(2+x)+3` we have

`abs(x-1)=y-> {(x-1=y),(x-1=-y):}`

1) From `x-1=y-> x = y+1` so
`y = abs(2+y+1)+3=abs(y+3)+3` but

`y+3=abs(y+3)+6` and
`pm 1=1+6/(y+3)` for `y ne -3` so the only possibility is
`-2=6/(y+3)-> y = -6->x = -5`

2)From `x-1=-y->x=-(y+1)` so
`y = abs(y-1)+3` but
`y-1=abs(y-1)+2` then
`pm1=1+2/(y-1)` for `y ne 1` so the only possibility is
`-2=2/(y-1)->y=0->x=1`

Substituting in the main equation we certify the solution: `x = -5`