1. **State the problem:** Solve the compound inequality $$2x - 1 \leq 3 + 1 < 2(x + 2)$$ and find all integral values of $x$ that satisfy it. 2. **Rewrite the inequality:** The compound inequality can be split into two parts: $$2x - 1 \leq 4$$ and $$4 < 2(x + 2)$$ 3. **Solve the first inequality:** $$2x - 1 \leq 4$$ Add 1 to both sides: $$2x \leq 5$$ Divide both sides by 2: $$x \leq \frac{5}{2}$$ 4. **Solve the second inequality:** $$4 < 2(x + 2)$$ Divide both sides by 2: $$2 < x + 2$$ Subtract 2 from both sides: $$0 < x$$ or equivalently $$x > 0$$ 5. **Combine the two inequalities:** $$0 < x \leq \frac{5}{2}$$ 6. **Find all integral values of $x$ that satisfy this:** The integers greater than 0 and less than or equal to $\frac{5}{2} = 2.5$ are: $$x = 1, 2$$ **Final answer:** The integral values of $x$ that satisfy the inequality are $1$ and $2$.