Final answer:
To find the solutions to r(x) > 0, analyze the intervals where the numerator and denominator are both positive or negative. Solve for x to find the critical point. Test values within each interval to determine if the function is positive or negative.
Step-by-step explanation:
To find the solutions to r(x) > 0, we need to determine when the function is positive. In order to do this, we analyze the intervals where the numerator (x-3) and denominator (x^5) are both positive or negative. When the numerator is positive and the denominator is negative, the overall fraction is negative. When the numerator is negative and the denominator is positive, the overall fraction is also negative. Therefore, the function is positive when either the numerator and denominator are both positive or both negative. We set these conditions equal to zero and solve for x to find the critical points: (x-3) = 0 and (x^5) = 0. Solving, we find that x = 3 is the only critical point. To determine if the function is positive or negative in each interval, we can test a value within each interval. For example, if we test x = 2, we find that the function is negative, indicating that the interval from negative infinity to 3 is negative. Similarly, if we test x = 4, we find that the function is positive, indicating that the interval from 3 to positive infinity is positive. Therefore, the solutions to r(x) > 0 are all values of x greater than 3.