Find all values of $x$ such that $\frac{x-4}{2x-3}$ is an integer.

Question

Find all values of $x$ such that $\frac{x-4}{2x-3}$ is an integer.

I tried many ways to do it, like by setting it to a fraction and what not, but it could just come for me, I hope you can help. Thanks a lot Math stack exchange. Thanks!

Still looking for an Anwser.

Answer 1

I.e. solve in integers $2x-3\mid x-4$. Since $2x-3$ is odd, this is equivalent to $2x-3\mid 2(x-4)=2x-8$, which is equivalent to $2x-3\mid (2x-8)-(2x-3)=-5$, which is equivalent to $2x-3\in\{-5,-1,1,5\}$.

Answer 2

$$\frac{x-4}{2x-3}=n\in\mathbb Z\implies x-4=2nx-3n\implies x=\frac{4-3n}{1-2n},n\in\mathbb Z$$