1. **Problem:** Solve for $x$ in the equation $\frac{1}{2}(x+6) - 3 = 4$. 2. **Formula and rules:** To solve linear equations, isolate $x$ by performing inverse operations step-by-step. 3. **Step 1:** Expand the left side: $$\frac{1}{2}(x+6) - 3 = 4 \implies \frac{1}{2}x + 3 - 3 = 4$$ 4. **Step 2:** Simplify the left side: $$\frac{1}{2}x + \cancel{3} - \cancel{3} = 4 \implies \frac{1}{2}x = 4$$ 5. **Step 3:** Multiply both sides by 2 to solve for $x$: $$2 \times \frac{1}{2}x = 2 \times 4 \implies \cancel{2} \times \frac{1}{\cancel{2}} x = 8 \implies x = 8$$ 6. **Answer:** $x = 8$. This completes the solution for the first problem.