Proof of irrationality of linear expression

Assume to the contrary that $\frac{2 + 3\sqrt{2}}{5} = r$ is a rational number. $2 + 3\sqrt{2} = 5r$ $3\sqrt{2} = 5r - 2$ $\sqrt{2} = \frac{5r - 2}{3}$ Since $r$ is rational, $\frac{5r - 2}{3}$ is also rational. But this implies $\sqrt{2}$ is rational, which is a contradiction because $\sqrt{2}$ is irrational. Hence, $\frac{2 + 3\sqrt{2}}{5}$ is an irrational number.