1. **State the problem:** Solve the equation $$|3x - 6| = 9$$ for real values of $x$. 2. **Recall the definition of absolute value:** For any real number $a$, $$|a| = b$$ means $$a = b$$ or $$a = -b$$ where $b \geq 0$. 3. **Apply this to our equation:** $$|3x - 6| = 9 \implies 3x - 6 = 9 \text{ or } 3x - 6 = -9$$ 4. **Solve each case separately:** - Case 1: $$3x - 6 = 9$$ Add 6 to both sides: $$3x - \cancel{6} + \cancel{6} = 9 + 6$$ $$3x = 15$$ Divide both sides by 3: $$\frac{3x}{\cancel{3}} = \frac{15}{\cancel{3}}$$ $$x = 5$$ - Case 2: $$3x - 6 = -9$$ Add 6 to both sides: $$3x - \cancel{6} + \cancel{6} = -9 + 6$$ $$3x = -3$$ Divide both sides by 3: $$\frac{3x}{\cancel{3}} = \frac{-3}{\cancel{3}}$$ $$x = -1$$ 5. **Final answer:** The solutions to the equation $$|3x - 6| = 9$$ are $$x = 5$$ and $$x = -1$$.