Solve the following inequality:

$$
\frac{x^{2}-7 x+12}{x^{2}-16 x+63}>0
$$
Our inequality can also be written as:

$$
(x-3)(x-4)(x-7)(x-9)>0
$$

This product is positive,

$1^{\circ}$. if all four factors are positive, that is, if

$$
x>9
$$

$2^{\circ}$. if all four factors are negative, that is, if

$$
x<3
$$

$3^{\circ}$. if two factors are positive and two are negative, that is, if

$$
4<x<7
$$

Thus, our function is positive if the value of $x$ is not between 3 and 4, or between 7 and 9. We can easily verify this solution by plotting the functions in the numerator and the denominator.