Final answer:
The solutions to the inequality r(x) > 0, where r(x) = (x-3)/(x+5), are when x < -5 and x > 3. These solutions correspond to when the numerator and denominator of the function have the same sign, resulting in a positive value for r(x).
Step-by-step explanation:
The question asks for the solutions to r(x) > 0 where the function r is given by r(x)=(x-3)/(x+5). To solve this inequality, we need to determine when the expression (x-3)/(x+5) is positive. This will occur when both the numerator and the denominator have the same sign, either both positive or both negative.
1. When x > 3, the numerator (x-3) is positive. For r(x) to be positive in this range, the denominator (x+5) must also be positive, which happens when x > -5. So, one part of the solution is x > 3.
2. When x < -5, both numerator and denominator are negative, which makes r(x) positive. Therefore, another part of the solution is x < -5.