Question

How do you solve `(x^2-x-12)/(x^2+4)<=0` using a sign chart?

Answer 1

The solution is `x in [-3,4]`

Explanation:

Factorise the numerator

`x^2-x-12=(x+3)(x-4)`

Therefore,

`(x^2-x-12)/(x^2+4)<=0`

`<=>`, `((x+3)(x-4))/(x^2+4)<=0`

`AA x in RR, (x^2+4)>0`

Let `f(x)=((x+3)(x-4))/(x^2+4)`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `-3` `color(white)(aaaaaaa)` `4` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+` `color(white)(aaaaaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaaaa)` `-` `color(white)(aa)` ###color(white)(aaaa)#`-` `color(white)(aa)` `0` `color(white)(aaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `+` `color(white)(aa)` `0` `color(white)(aaa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+`

Therefore,

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

graph{(x^2-x-12)/(x^2+4) [-10, 10, -5, 5]}