Question

How do you solve the inequality: `((8-x)^3(8x+1))/(x^3 - 1) > 0`?

Answer 1

The solution is `x in (-oo,-1/8) uu(1,8)`

Explanation:

`"Reminder"`

`x^3-1=(x-1)(x^2+x+1)`

Therefore,

`((8-x)^3(8x+1))/(x^3-1)=((8-x)^3(8x+1))/((x-1)(x^2+x+1))`

The sign of `(x^2+x+1) >0` as the discriminant is `Delta<0`

Let `f(x)=((8-x)^3(8x+1))/((x-1)(x^2+x+1))`

`"Let's build the sign chart"`

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `-1/8` `color(white)(aaaaaaa)` `1` `color(white)(aaaaaaa)` `8` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `8x+1` `color(white)(aaaa)` `-` `color(white)(aaaa)` `0` `color(white)(aaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-1` `color(white)(aaaaa)` `-` `color(white)(aaaa)` ###color(white)(aaaa)#`-` `color(white)(aa)` `||` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `(8-x)^3` `color(white)(aaa)` `+` `color(white)(aaaa)` ###color(white)(aaaa)#`+` `color(white)(aaaaa)` `+` `color(white)(aa)` `0` `color(white)(a)` `-`

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

Therefore,

`f(x)>0` when `x in (-oo,-1/8) uu(1,8)`

graph{(8-x)^3(8x+1)/(x^3-1) [-46.23, 46.24, -23.1, 23.15]}