Unity Games is a company that sells video games. Its revenue last year was modelled by the function g(x)=-4x4+6x(4x-1)-4 and its revenue this year is modelled by the functionf(x)=-3x2(x2-8)-6x -5where x is the number of video games sold in thousands and 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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To determine the range of values for the number of video games sold that makes the revenue this year greater than the revenue last year, we need to compare the two revenue functions and find the values of x for which f(x) is greater than g(x).
First, let's compare the two functions:
g(x) = -4x^4 + 6x(4x-1) - 4
f(x) = -3x^2(x^2-8) - 6x - 5
To find the values of x for which f(x) is greater than g(x), we need to set up the inequality:
f(x) > g(x)
Substitute the values of f(x) and g(x) into the inequality:
-3x^2(x^2-8) - 6x - 5 > -4x^4 + 6x(4x-1) - 4
Simplify the inequality:
-3x^4 + 24x^2 - 6x - 5 > -4x^4 + 24x^2 - 6x - 4
Combine like terms:
x^4 > 1
Take the square root of both sides:
x^2 > 1
Take the square root again:
x > ±1
So, the range of values for the number of video games sold that makes the revenue this year greater than the revenue last year is x > 1 (in thousands).
Therefore, if the number of video games sold (in thousands) is greater than 1, the revenue this year will be greater than the revenue last year.