Question

How do you solve and write the following in interval notation: `(1-x)/(x-9) >=0`?

Answer 1

The solution is `x in [1, 9[`

Explanation:

Let `f(x)=(1-x)/(x-9)`

The domain of `f(x)` is `D_f(x)=RR-{9}`

We build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `1` `color(white)(aaaaaaaa)` `9` `color(white)(aaaaaaa)` `+oo`

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

`color(white)(aaaa)` `x-9` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)>=0`, when `x in [1, 9[`