To determine when a rational function is less than or equal to a specific value, we can set up and solve a rational inequality. For example, given the function $R(x) = \frac{x+3}{x-5}$, to find the values of $x$ that make the function less than or equal to $0$, we solve the inequality $\frac{x+3}{x-5} \leq 0$. First, find the zero partition numbers by setting the numerator and denominator to zero: $x+3=0 \implies x=-3$ and $x-5=0 \implies x=5$. These partition numbers divide the number line into three intervals: $(-\infty, -3)$, $(-3, 5)$, and $(5, \infty)$. Testing values in each interval reveals that the quotient is negative in the interval $(-3, 5)$. Since the inequality requires the expression to be less than or equal to $0$, the solution must include the values where the quotient is negative or zero. The partition number $x=5$ must be excluded because it makes the denominator zero (and thus the function undefined), while $x=-3$ is included because it makes the numerator, and the entire function, equal to $0$. Therefore, written in interval notation, the solution is $[-3, 5)$.