1. **State the problem:** Find the value of $x$ that satisfies the equation $$\frac{1}{3}(5x+3) - \frac{1}{6}(x+6) = 6.$$\n\n2. **Write down the equation clearly:** $$\frac{1}{3}(5x+3) - \frac{1}{6}(x+6) = 6.$$\n\n3. **Distribute the fractions:**\n$$\frac{1}{3} \times 5x = \frac{5x}{3}, \quad \frac{1}{3} \times 3 = 1,$$\n$$\frac{1}{6} \times x = \frac{x}{6}, \quad \frac{1}{6} \times 6 = 1.$$\nSo the equation becomes $$\frac{5x}{3} + 1 - \frac{x}{6} - 1 = 6.$$\n\n4. **Simplify the constants:** $$1 - 1 = 0,$$ so the equation reduces to $$\frac{5x}{3} - \frac{x}{6} = 6.$$\n\n5. **Find a common denominator to combine terms:** The common denominator of 3 and 6 is 6. Rewrite $$\frac{5x}{3} = \frac{10x}{6}.$$\nSo the equation is $$\frac{10x}{6} - \frac{x}{6} = 6.$$\n\n6. **Combine the fractions:** $$\frac{10x - x}{6} = \frac{9x}{6} = 6.$$\n\n7. **Solve for $x$:** Multiply both sides by 6 to clear the denominator: $$9x = 6 \times 6 = 36.$$\nDivide both sides by 9: $$x = \frac{36}{9} = 4.$$\n\n**Final answer:** $$\boxed{4}.$$