Question

How do you solve `(x^2-4)/(3-x)>=0` using a sign chart?

Answer 1

The answer is `x in] -oo,-2 ] uu [2, 3[`

Explanation:

Let `f(x)=(x^2-4)/(3-x)=((x-2)(x+2))/(3-x)`

The domain is `D_f=RR-{3}`

Let's do the sign chart

`color(white)(aaaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `-2` `color(white)(aaaaa)` `2` `color(white)(aaaaa)` `3` `color(white)(aaaaa)` `+oo`

`color(white)(aaaaa)` `x+2` `color(white)(aaaaa)` `-` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaaa)` `x-2` `color(white)(aaaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaaa)` `3-x` `color(white)(aaaaa)` `+` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaa)` `-`

`color(white)(aaaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaa)` `-`

Therefore, `f(x)>=0`

when `x in] -oo,-2 ] uu [2, 3[`

graph{(x^2-4)/(3-x) [-34.51, 30.44, -18.83, 13.64]}