## Discrete Mathematics with Applications 4th Edition

By definition, a number is rational if it is a ratio of two integers, of which the second (the denominator) is not zero. Since $m$ and $n$ are integers, $5m$, $12n$, and $4n$ must all be integers by the closure of the integers under multiplication. From this we conclude that $5m-12n$ is an integer by the closure of the integers under subtraction. Finally, $4n\ne0$ by the zero product property, since $4\ne0$ trivially and we are told that $n\ne0$. Therefore, $\frac{5m-12n}{4n}$ is a ratio of two integers of which the second is not zero, so it is a rational number by definition.