Question

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

Answer 1

The answer is `x in ] -oo, 0] uu ] 9,+ oo[`

Explanation:

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

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

Let's do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `0` `color(white)(aaaa)` ###color(white)(aaaa)#`9` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaa)` `-` `color(white)(aaaaaa)` `+` `color(white)(a)` ###color(white)(a)#`∥` `color(white)(aa)` `+`

`color(white)(aaaa)` `x-9` `color(white)(aaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(a)` ###color(white)(a)#`∥` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `+` `color(white)(aaaaaa)` `-` `color(white)(a)` ###color(white)(a)#`∥` `color(white)(aa)` `+`

Therefore,

`f(x)>=0` when `x in ] -oo, 0] uu ] 9,+ oo[`