Solve the inequality writing it as an interval

label Algebra
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Solve the inequality

x^3 + 5x^2 >or equal to 4x + 20

Write your answer as an interval or union of intervals.

Please note that the sign is supposed to have a line under it making it 'or equal to.'

Oct 22nd, 2015

Thank you for the opportunity to help you with your question!

x^3 + 5x^2 - 4x - 20 >=0

First, make this into an equation:

x^3 + 5x^2 - 4x - 20 = 0

solve for x by factoring by grouping:

(x^3 + 5x^2) - (4x + 20) = 0

factor out common term

x^2(x+5) - 4(x+5) = 0

Factor out the (x+5)

(x+5)(x^2-4) = 0

the x^2-4 is a difference of two squares and thus can be factored more

(x+5)(x+2)(x-2) = 0

set each term to zero and solve for x

x+5 = 0 OR x+2 = 0 OR x-2 = 0
x = -5 OR x = -2 OR x = 2

plot these on a number line:

______________________________________...
-5 -2 2

pick values in each interval to see if they satisfy the inequality:

-10:
(-10)^3 + 5(-10)^2 - 4(-10) - 20 >=0
-1000 + 500 + 40 - 20 >= 0 FALSE

-1:
(-1)^3 + 5(-1)^2 - 4(-1) - 20 >=0
-1 + 5 + 4 - 20 <=0 FALSE

0:
-20 >=0 FALSE

3:
3^3 + 5(3)^2 - 4(3) - 20 >= 0 TRUE 

therefore are solutions is true when the value of x lies between -5 and -2 and when it is greater than or equal to 2

as interval notation:
[-5, -2] U [2,inf)

Let me know incase you need any further help ! Thanks :)
Oct 22nd, 2015

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