Question

How do you solve `x/(x-2)>=0`?

Answer 1

The solution is `x in (-oo, 0] uu(2, +oo)`

Explanation:

Let `f(x)=x/(x-2)`

Build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaaa)` `0` `color(white)(aaaaaaaa)` `2` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aaaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x-2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` ###color(white)(aaaaa)#`-` `color(white)(aa)` `||` `color(white)(aa)` `+`

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

Therefore,

`f(x)>=0` when ##

graph{x/(x-2) [-10, 10, -5, 5]}

Answer 2

`(-oo, 0]` U `(2, +oo)`

Explanation:

`x /(x - 2)≥0`

`x /(x - 2)≥0" : is true if" {("either", x ≥0 and x - 2 > 0),("or",x ≤ 0 and x - 2 < 0):}`

`x ≥0 and x - 2 > 0`
`x > 2`

`x ≤ 0 and x - 2 < 0`
`x ≤ 0`

Answer: `x ≤ 0` OR `x > 2`
In interval notation: `(-oo, 0]` U `(2, +oo)`