Question

How do you solve and write the following in interval notation:`((x+8)(x-5) )/ (x-1) ≥ 0`?

Answer 1

The solution is `x in [-8,1 [uu [5, +oo [`

Explanation:

Let `f(x)=((x+8)(x-5))/(x-1)`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaa)` `-oo` `color(white)(aaaa)` `-8` `color(white)(aaaaaaa)` `1` `color(white)(aaaaaaaa)` `5` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `x+8` `color(white)(aaaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `+`

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

`color(white)(aaaa)` `x-5` `color(white)(aaaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `+`

Therefore,

`f(x)>=0` when `x in [-8,1 [uu [5, +oo [`