Question

How do you solve and write the following in interval notation: `(4-x) /( x-8)>=0`?

Answer 1

The answer is `x in [ 4, -8 [`

Explanation:

Let `f(x)=(4-x)/(x-8)`

The domain of `f(x)` is `D_f(x)=RR-{8}`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `4` `color(white)(aaaaaaaa)` `8` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `4-x` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `color(red)(||)` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `x-8` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `color(red)(||)` `color(white)(aaaa)` `+`

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

Therefore,

`f(x>=0)` when `x in [ 4, -8 [`