1. **State the problem:** Solve for $x$ in the equation $$3|2x + 1| - 3 = 9.$$\n\n2. **Isolate the absolute value expression:** Add 3 to both sides:\n$$3|2x + 1| - 3 + 3 = 9 + 3$$\n$$3|2x + 1| = 12$$\n\n3. **Divide both sides by 3:**\n$$\cancel{3}|2x + 1| = \frac{12}{\cancel{3}}$$\n$$|2x + 1| = 4$$\n\n4. **Recall the definition of absolute value:**\nIf $|A| = B$, then $A = B$ or $A = -B$. Here, $A = 2x + 1$ and $B = 4$.\n\n5. **Set up two equations:**\n$$2x + 1 = 4 \quad \text{or} \quad 2x + 1 = -4$$\n\n6. **Solve each equation:**\n- For $2x + 1 = 4$:\n$$2x = 4 - 1$$\n$$2x = 3$$\n$$x = \frac{3}{2}$$\n\n- For $2x + 1 = -4$:\n$$2x = -4 - 1$$\n$$2x = -5$$\n$$x = \frac{-5}{2}$$\n\n**Final answer:** $$x = \frac{3}{2} \text{ or } x = \frac{-5}{2}.$$