How do you solve the inequality: 9x - x^2<=20?

Aug 6, 2015

$x \ge 5$ or $x \le 4$

Explanation:

$9 x - {x}^{2} \le 20$

is equivalent to
$\textcolor{w h i t e}{\text{XXXX}}$$0 \le {x}^{2} + 9 x + 20$
or, after factoring,
$\textcolor{w h i t e}{\text{XXXX}}$$\left(x - 5\right) \left(x - 4\right) \ge 0$

$\left(x - 5\right) \left(x - 4\right)$ will be $\ge 0$
if both terms are $\ge 0$
$\textcolor{w h i t e}{\text{XXXX}}$(that is if $x \ge 5$)
or if both terms are $\le 0$
$\textcolor{w h i t e}{\text{XXXX}}$(that is if $x \le 4$)

Aug 7, 2015

Solve $9 x - {x}^{2} \le 20$

Ans: (-infinity, 4] and [5, infinity)

Explanation:

Standard form: f(x) = - x^2 + 9x - 20 <= 0
First solve the quadratic equation y = - x^2 + 9x - 20 = 0. Factor pairs of ac = 20 -> (2, 10)(4, 5). This sum is 9 = b (a < 0)
The 2 real roots are: 4 and 5.
Since a < 0. the parabola opens downward, between the 2 real roots 4 and 5, f(x) > 0, and f(x) < 0 outside this interval.
The 2 critical points 4 and 5 are included in the solution set.
Answer by half closed intervals: (-infinity, 4] and [5, +infinity)
graph{-x^2 + 9x - 20 [-10, 10, -5, 5]}