1. **State the problem:** Solve the absolute value equation $$3|x - 4| = 33$$ for $x$. 2. **Isolate the absolute value:** Divide both sides by 3 to get $$|x - 4| = \frac{33}{3} = 11$$. 3. **Recall the definition of absolute value:** For $|A| = B$ where $B \geq 0$, the solutions are $A = B$ or $A = -B$. 4. **Apply this to our equation:** $$x - 4 = 11 \quad \text{or} \quad x - 4 = -11$$ 5. **Solve each equation:** - For $x - 4 = 11$, add 4 to both sides: $$x = 11 + 4 = 15$$ - For $x - 4 = -11$, add 4 to both sides: $$x = -11 + 4 = -7$$ 6. **Write the solutions in order:** The smallest solution is $-7$, and the other is $15$. **Final answer:** $$x = -7, 15$$