1. **State the problem:** Simplify the expression $\left(\frac{1}{2}x - 3\right) + \left(\frac{1}{6}x + 1\right)$. 2. **Write the expression:** $$\frac{1}{2}x - 3 + \frac{1}{6}x + 1$$ 3. **Group like terms:** $$\left(\frac{1}{2}x + \frac{1}{6}x\right) + (-3 + 1)$$ 4. **Find common denominator for $x$ terms:** $$\frac{1}{2}x = \frac{3}{6}x$$ 5. **Add $x$ terms:** $$\frac{3}{6}x + \frac{1}{6}x = \frac{3+1}{6}x = \frac{4}{6}x$$ 6. **Simplify fraction:** $$\frac{4}{6}x = \frac{\cancel{2} \times 2}{\cancel{2} \times 3}x = \frac{2}{3}x$$ 7. **Add constants:** $$-3 + 1 = -2$$ 8. **Final simplified expression:** $$\frac{2}{3}x - 2$$