The inverse of a relation is obtained by swapping the coordinates of each ordered pair in the set. For the relation $S = \{(8,7), (3,4), (-1,-8), (-3,-9)\}$, we will swap the $x$- and $y$-coordinates of each pair.

Thus, the inverse relation $S^{-1}$ is:

$S^{-1} = \{(7,8), (4,3), (-8,-1), (-9,-3)\}$

Let me know if you'd like further details on how this works, or if you have any other questions!

Here are 5 related questions:

1. How do you determine if a relation is a function?
2. Can a relation and its inverse both be functions?
3. What is the graphical interpretation of an inverse relation?
4. How do you find the inverse of a function algebraically?
5. What is the difference between a one-to-one function and a many-to-one function?

Tip: When finding the inverse of a function or relation, always ensure you swap the x- and y-values carefully, especially for negative numbers!