Question

How do you solve `((x+7)(x-3))/(x-1)>=0`?

Answer 1

The answer is `x in [-7,1 [ uu [3,+oo[`

Explanation:

Let `f(x)=((x+7)(x-3))/(x-1)`

The domain of `f(x)` is `D_f(x)=RR-{1}`

Now we can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-7` `color(white)(aaaaaaa)` `1` `color(white)(aaaaaa)` `3` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `x+7` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

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

`color(white)(aaaa)` `x-3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)>=0` when `x in [-7,1 [ uu [3,+oo[`