1. **State the problem:** Solve the inequality $|3|9 - 6x|| < 45$ for $x$. 2. **Simplify the expression inside the absolute value:** Since $3|9 - 6x|$ means $3$ times the absolute value of $(9 - 6x)$, rewrite the inequality as: $$3|9 - 6x| < 45$$ 3. **Isolate the absolute value:** Divide both sides by 3: $$\frac{3|9 - 6x|}{\cancel{3}} < \frac{45}{\cancel{3}}$$ which simplifies to: $$|9 - 6x| < 15$$ 4. **Solve the absolute value inequality:** Recall that $|A| < B$ means $-B < A < B$. So: $$-15 < 9 - 6x < 15$$ 5. **Solve the compound inequality:** Subtract 9 from all parts: $$-15 - 9 < 9 - 6x - 9 < 15 - 9$$ which is: $$-24 < -6x < 6$$ 6. **Divide all parts by -6, remembering to reverse inequality signs because dividing by a negative number:** $$\frac{-24}{-6} > x > \frac{6}{-6}$$ which simplifies to: $$4 > x > -1$$ or equivalently: $$-1 < x < 4$$ **Final answer:** $$\boxed{-1 < x < 4}$$