Question

How do you solve `\frac { x + 3} { x - 5} < 1`?

Answer 1

The solution is `x in (-oo,5)`

Explanation:

Let's rewrite and simplify the inequality

We cannot do crossing over

`(x+3)/(x-5)<1`

`(x+3)/(x-5)-1<0`

`(x+3-x+5)/(x-5)<0`

`8/(x-5)<0`

Let `f(x)=8/(x-5)`

This is a simple sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaaaaaa)` `5` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `(x-5)` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `||` `color(white)(aaaa)` `+`

Therefore,

`f(x)<0`, when `x in (-oo,5)`
graph{(x+3)/(x-5)-1 [-22.81, 22.8, -11.4, 11.42]}