Question

How do you solve `x^2-6x+9>16` using a sign chart?

Answer 1

The solution is `x in ]-oo,-1[uu]7,+oo[`

Explanation:

Let's rewrite the inequality and factorise

`x^2-6x+9-16>0`

`x^2-6x-7>0`

`(x-7)(x+1)>0`

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

We build the sign chart

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

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

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

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

Therefore,

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