1. **State the problem:** Solve the equation $$\frac{3}{2} (2 - 6x) + 1 = 3 - 5x.$$\n\n2. **Use the distributive property:** Multiply $$\frac{3}{2}$$ by each term inside the parentheses.\n$$\frac{3}{2} \times 2 = 3$$\n$$\frac{3}{2} \times (-6x) = -9x$$\nSo the equation becomes:\n$$3 - 9x + 1 = 3 - 5x.$$\n\n3. **Combine like terms on the left side:**\n$$3 + 1 = 4,$$\nso the equation is now:\n$$4 - 9x = 3 - 5x.$$\n\n4. **Bring all terms involving $$x$$ to one side and constants to the other:**\nAdd $$9x$$ to both sides:\n$$4 - \cancel{9x} + 9x = 3 - 5x + 9x$$\nwhich simplifies to:\n$$4 = 3 + 4x.$$\n\n5. **Isolate $$x$$:**\nSubtract 3 from both sides:\n$$4 - 3 = 3 - 3 + 4x$$\n$$1 = 4x.$$\n\n6. **Solve for $$x$$ by dividing both sides by 4:**\n$$\frac{1}{\cancel{4}} = \frac{4x}{\cancel{4}}$$\n$$x = \frac{1}{4}.$$\n\n**Final answer:** $$x = \frac{1}{4}.$$