Question

What is the value of `x` given that `(x+3)/(x+7)>3`?

Answer 1

The solution is `x in (-9,-7)`

Explanation:

You cannot do crossing over

The inequality is

`(x+3)/(x+7)>3`

`=>`, `(x+3)/(x+7)-3>0`

`=>`, `(x+3-3(x+7))/(x+7)`

`=>`, `(x+3-3x-21)/(x+7)>0`

`=>`, `(-2x-18)/(x+7)>0`

`=>`, `(2(x+9))/(x+7)<0`

Let `f(x)=(2(x+9))/(x+7)`

Let's build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-9` `color(white)(aaaa)` `-7` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+9` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+7` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)<0` when `x in (-9,-7)`

graph{(x+3)/(x+7)-3 [-26.83, 9.2, -8.96, 9.06]}