Question

How do you solve `(x+1)/(x+3)<2`?

Answer 1

There are two cases to investigate

Explanation:

(1)
`x> -3->x+3>0`
We may multiply both sides by `x+3` and the `<` sign stays:
`x+1<2(x+3)->x+1<2x+6->` subtract `x` and `6`
`cancelx-cancelx+1-6<2x-x+cancel6-cancel6->`
`-5 < x->x> -5`
This, with the first assumption, leads to `x> -3`
(2)
`x<-3->x+3<0`
Multiply by `x+3` but the `<`sign becomes `>`
`x+1>2x+6->` subtract `x` and `6`
`cancelx-cancelx+1-6>2x-x+cancel6-cancel6->`
`-5 > x->x< -5`

Combined: `x< -5orx> -3`

graph{(x+1)/(x+3) [-10, 10, -5, 5]}