1. **State the problem:** Solve the equation $$\frac{1}{2} + \frac{2}{3} = \frac{1}{x}$$ and find any non-permissible values for $x$. 2. **Identify non-permissible values:** Since $x$ is in the denominator, $x \neq 0$ to avoid division by zero. 3. **Find a common denominator on the left side:** The denominators are 2 and 3, so the common denominator is 6. 4. **Rewrite the left side:** $$\frac{1}{2} = \frac{3}{6}, \quad \frac{2}{3} = \frac{4}{6}$$ 5. **Add the fractions:** $$\frac{3}{6} + \frac{4}{6} = \frac{7}{6}$$ 6. **Set the equation:** $$\frac{7}{6} = \frac{1}{x}$$ 7. **Cross multiply:** $$7x = 6$$ 8. **Solve for $x$:** $$x = \frac{6}{7}$$ 9. **Check non-permissible values:** $x = \frac{6}{7} \neq 0$, so it is valid. **Final answer:** $$x = \frac{6}{7}$$