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

Answer 1

Solve #9x - x^2 <= 20#

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

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]}
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Answer 2

#x>=5# or #x<=4#

#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#)
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Answer 3

To solve the inequality (9x - x^2 \leq 20), you first rearrange it into standard quadratic form by moving all terms to one side: (x^2 - 9x + 20 \geq 0). Then, factor the quadratic expression: ((x - 5)(x - 4) \geq 0). Now, you can find the critical points by setting each factor equal to zero and solving for x: (x - 5 = 0) or (x - 4 = 0), which gives (x = 5) or (x = 4). Next, plot these critical points on a number line. Since the inequality is greater than or equal to zero, you want to determine where the quadratic expression is positive or zero. This occurs when (x) is less than or equal to 4, or when (x) is greater than or equal to 5. Therefore, the solution to the inequality is (x \leq 4) or (x \geq 5).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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