Question

How do you solve `x/(x-3)>0` using a sign chart?

Answer 1

`x<0` or `x>3`

Explanation:

A rational function of the form `(f(x))/(g(x))` will change signs when exactly one of `f(x)` and `g(x)` changes signs.

Thus, the function `x/(x-3)` will change signs when exactly one of `x` and `x-3` changes signs. `x` changes signs when `x=0` and `x-3` changes signs when `x=3`.

When it changes signs can be solved by equating it to `0` or finding when it is undefined. These are the only values where it can change signs. In order to know if it actually changes signs, look at two values, one larger and one smaller, and see if it actually changes signs.
For example, for `(x-3)^2(x-1)`, the only value which it can change signs is `x=3` and `x=1`. However, if you check the nearby values for `x=3`, you will find that the function still stays positive and does not change signs.

Thus, we can create a sign diagram for `x/(x-3)`. This function will change signs at `x=0` and `x=3`. Now, these two values divide the function domain into three areas, `x<0`, `0< x<3`, and `x>3`. Check one value inside each area to determine the sign of each of the three areas. From this, we can create the sign diagram:

`color(white)(--)"positive"color(white)(----)"negative"color(white)(-----)"positive"`
`larr---- | -------- | ------>`
`color(white)(-----)x=0color(white)(------)x=3`

From this sign diagram, we can see that `x/(x-3)>0` when `x<0` and `x>3`. (Note that at the points `x=0` the function is `0`, and at the point `x=3` the function is undefined.)

We can draw a graph to verify this answer:
graph{x/(x-3) [-10, 10, -5, 5]}