Final answer:
To find the solutions to r(x) >= 0, we need to find the values of x that make r(x) greater than or equal to zero. By finding the critical points and testing the intervals between them, we can determine that the solution is x < 3. This corresponds to option a) x < 0.
Step-by-step explanation:
To find the solutions to r(x) >= 0, we need to find the values of x that make r(x) greater than or equal to zero. In this case, we have the function r(x) = (x-3)/x^5. To find the solutions, we need to determine the values of x that make r(x) greater than or equal to zero. We can do this by finding the critical points and testing the intervals between them.
First, we need to find the critical points by setting r(x) equal to zero and solving for x. Setting r(x) = 0, we get (x-3)/x^5 = 0. Since a fraction is equal to zero only when the numerator is equal to zero, we have x-3 = 0. Solving for x, we find x = 3. Therefore, x = 3 is a critical point.
Since r(x) is a rational function, it is undefined at x = 0. Therefore, x = 0 is also a critical point. However, we cannot include x = 0 in our solution because it would make r(x) undefined. So the only critical point we consider is x = 3.
Now, we can test the intervals between the critical point x = 3 and negative infinity by plugging in values into r(x) to see if r(x) is positive or negative. For any value of x less than 3, r(x) will be negative because the numerator (x-3) will be negative and the denominator (x^5) will be positive. Therefore, the solution is x < 3. This corresponds to option a) x < 0.
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