1. **State the problem:** Solve the inequality $|x + 29| - 1 < 7$. 2. **Isolate the absolute value:** Add 1 to both sides: $$|x + 29| - 1 + 1 < 7 + 1$$ $$|x + 29| < 8$$ 3. **Recall the rule for absolute value inequalities:** For $|A| < B$ where $B > 0$, the solution is: $$-B < A < B$$ 4. **Apply the rule:** $$-8 < x + 29 < 8$$ 5. **Solve the compound inequality:** Subtract 29 from all parts: $$-8 - 29 < x + 29 - 29 < 8 - 29$$ $$-37 < x < -21$$ 6. **Final answer:** The solution set is all $x$ such that $$x \in (-37, -21)$$