[tex]\it \dfrac{2x+3}{x-4}\in\mathbb{Z}\Rightarrow\dfrac{2x+3+8-8}{x-4} \in\mathbb{Z}\Rightarrow \dfrac{2(x-4)}{x-4}+\dfrac{11}{x-4}\in\mathbb{Z}\Rightarrow 2+\dfrac{11}{x-4} \in\mathbb{Z}\\ \\ \\ \Rightarrow \dfrac{11}{x-4}\in\mathbb{Z} \Rightarrow x-4|11\Rightarrow x-4\in\{\pm1,\pm11\}\Rightarrow x-4\in\{-11,\ -1,\ 1,\ 11\}|_{+4}\Rightarrow\\ \\ \\ \Rightarrow x\in\{-7,\ 3,\ 5,\ 15\}[/tex]
