1. **State the problem:** Solve the inequality $|2b - 1| \leq 3$. 2. **Recall the definition of absolute value inequality:** For any expression $x$, $|x| \leq a$ means $-a \leq x \leq a$. 3. **Apply this to our problem:** $$-3 \leq 2b - 1 \leq 3$$ 4. **Solve the compound inequality:** Add 1 to all parts: $$-3 + 1 \leq 2b - 1 + 1 \leq 3 + 1$$ $$-2 \leq 2b \leq 4$$ Divide all parts by 2: $$\frac{-2}{2} \leq \frac{2b}{2} \leq \frac{4}{2}$$ $$-1 \leq b \leq 2$$ 5. **Interpretation:** The solution set for $b$ is all real numbers between $-1$ and $2$, inclusive. **Final answer:** $$-1 \leq b \leq 2$$