Question

How do you solve `(x+10)/(x+3)<2`?

Answer 1

The solution is `x in (-3,4)`

Explanation:

We cannot do crossing over

`(x+10)/(x+3)<2`

Let's rewrite the inequality

`(x+10)/(x+3)-2<0`

`((x+10)-2(x+3))/(x+3)<0`

`((x+10-2x-6))/(x+3)<0`

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

Let `f(x)=(4-x)/(x+3)`

We build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-3` `color(white)(aaaaaaaa)` `4` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `4-x` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aa)` `0` `color(white)(aa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aa)` `0` `color(white)(aa)` `-`

Therefore,

`f(x)<0` when `x in (-3,4)`

graph{(x+10)/(x+3)-2 [-16.02, 16.02, -8.01, 8.02]}