Polynomial Equations and Inequalities

Question

Unity Games is a company that sells video games. Its revenue last year was modelled by the function g(x)=-4x^4+6x(4x-1)-4 and its revenue this year is modelled by the function f(x)=-3x^2((x^2)-8)-6x-5 where x is the number of video games sold in the thousands and the revenue is in millions of dollars. Determine the range of values for the number of video games sold that makes the revenue this year greater than the revenue last year. You must use a full algebraic solution.

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Answer 1

Last year: g(x)=-4x⁴+6x(4x-1)-4; this year: f(x)=-3x²(x²-8)-6x-5,

Last year: g(x)=-4x⁴+24x²-6x-4; this year f(x)=-3x⁴+24x²-6x-5.

We need to know when f(x)>g(x), that is, f(x)-g(x)>0.

Therefore: -3x⁴+24x²-6x-5-(-4x⁴+24x²-6x-4)>0,

-3x⁴+24x²-6x-5+4x⁴-24x²+6x+4>0,

x⁴-1>0, x⁴>1⇒x>1, because x has to be positive since it's a quantity of videos. So this year's revenue exceeds last year's provided more than 1,000 videos are sold (revenue exceeds $10M).

[However, it should be noted that the net revenue for the two years can be a loss if the quantity of videos exceeds a certain amount. For last year, this would have been x=2.268 (2,268 videos) and for this year x=2.647 (2,647 videos). Nevertheless, this year's revenue still exceeds last year's (and losses are not as great).]