[tex]\frac{x+2}{\sqrt{2}}=\frac{x+3}{\sqrt{3}} \\\\ \frac{\sqrt{2}(x+2)}{2}=\frac{\sqrt{3}(x+3)}{3}\\\\ 3\sqrt{2}(x+2)=2\sqrt{3}(x+3) \\\\ 3x\sqrt{2}+6\sqrt{2}=2x\sqrt{3}+6\sqrt{3} \\\\ 3x\sqrt{2}-2x\sqrt{3}= 6\sqrt{3}-6\sqrt{2} \\\\ x(3\sqrt{2}-2\sqrt{3})=6(\sqrt{3}-\sqrt{2}) \\\\ x= \frac{6(\sqrt{3}-\sqrt{2})}{3\sqrt{2}-2\sqrt{3}} \\\\ x= \frac{6(3\sqrt{2}+2\sqrt{3})(\sqrt{3}-\sqrt{2})}{18-12} \\\\ x= \frac{\not 6(3\sqrt{6}-6+6-2\sqrt{6})}{\not 6} \\\\ \boxed{x= \sqrt{6}}[/tex]
