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

2 Answers
Aug 6, 2015

Answer:

#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#)

Aug 7, 2015

Answer:

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]}