Question

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

Answer 1

`x>=5` or `x<=4`

Explanation:

`9x-x^2<=20`

is equivalent to
`color(white)("XXXX")` `0 <= x^2+9x+20`
or, after factoring,
`color(white)("XXXX")` `(x-5)(x-4)>=0`

`(x-5)(x-4)` will be `>=0`
if both terms are `>=0`
`color(white)("XXXX")`(that is if `x>=5`)
or if both terms are `<=0`
`color(white)("XXXX")`(that is if `x<=4`)

Answer 2

Solve `9x - x^2 <= 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]}