Question

How do you solve `x^2-3x>54` using a sign chart?

Answer 1

The answer is `x in ] -oo,-6 [ uu ] 9, oo [`

Explanation:

Let's rewrite the inequality

`x^2-3x-54>0`

Let's factorise

`x^2-3x-54=(x+6)(x-9)`

and let `f(x)=x^2-3x-54`

Now we can make the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-6` `color(white)(aaaa)` `9` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+6` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-9` `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,-6 [ uu ] 9, oo [`