Question

How do you solve the inequality `(x^2-2x-24)/(x^2-8x-20)>=0`?

Answer 1

The solution is `x in (-oo,-4] uu (-2, 6] uu (10, +oo)`

Explanation:

Factorise the inequality

`(x^2-2x-24)/(x^2-8x-20)=((x+4)(x-6))/((x+2)(x-10))>=0`

Let `f(x)=((x+4)(x-6))/((x+2)(x-10))`

Perform a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-4` `color(white)(aaaa)` `-2` `color(white)(aaaaa)` `6` `color(white)(aaaa)` `10` `color(white)(aaaa)` `+oo`##

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

`color(white)(aaaa)` `x+2` `color(white)(aaaaa)` `-` `color(white)(aa)` ###color(white)(aaaa)#`-` `color(white)(a)` `||` `color(white)(a)` `+` `color(white)(aaa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-6` `color(white)(aaaaa)` `-` `color(white)(aa)` ###color(white)(aaaa)#`-` `color(white)(a)` ###color(white)(aa)#`-` `color(white)(a)` `0` `color(white)(a)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-10` `color(white)(aaaa)` `-` `color(white)(aa)` ###color(white)(aaaa)#`-` `color(white)(a)` ###color(white)(aa)#`-` `color(white)(a)` ###color(white)(aa)#`-` `color(white)(a)` `||` `color(white)(a)` `+`

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

Therefore,

`f(x)>=0` when `x in (-oo,-4] uu (-2, 6] uu (10, +oo)`

graph{(x^2-2x-24)/(x^2-8x-20) [-25.65, 25.66, -12.83, 12.84]}