Question

How do you solve the inequality `x^2 - 9x> -18`?

Answer 1

The solutions are `x in (-oo,3) uu(6, +oo)`

Explanation:

The inequality is

`x^2-9x > -18`

`x^2-9x +18 > 0`

`(x-3)(x-6) >0`

Let `f(x)=(x-3)(x-6)`

Let's build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `3` `color(white)(aaaaa)` `6` `color(white)(aaaaaa)` `+oo`

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

`color(white)(aaaa)` `x-6` `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,3) uu(6, +oo)`

graph{x^2-9x+18 [-4.29, 15.71, -3.96, 6.04]}

Answer 2

(-inf., 3) and (6, +inf.)

Explanation:

#x^2 - 9x > - 18$ #f(x) = x^2 - 9x + 18 > 0#
Find 2 real roots, that have same sign(ac > 0), knowing the sum (- b = 9) and the product (c = 18). They are 6 and 3.
The graph of f(x) is an upward parabola (a > 0).
Inside the interval (3, 6) --> f(x) < 0 as the graph is below the x-axis.
Outside the interval (3, 6) --> f (x )> 0. Therefor, the solution set is the 2 open intervals:
(- inf., 3) and (6, +inf.)