1. **State the problem:** Solve the inequality $$-3 < 2(x + 1) < 7$$. 2. **Understand the inequality:** This is a compound inequality meaning both inequalities must be true simultaneously. 3. **Isolate the variable term:** Divide all parts of the inequality by 2 to simplify. $$-3 < 2(x + 1) < 7$$ Divide by 2: $$\frac{-3}{2} < \cancel{\frac{2}{2}}(x + 1) < \frac{7}{2}$$ which simplifies to $$-\frac{3}{2} < x + 1 < \frac{7}{2}$$ 4. **Solve for $x$:** Subtract 1 from all parts. $$-\frac{3}{2} - 1 < x + 1 - 1 < \frac{7}{2} - 1$$ Simplify: $$-\frac{3}{2} - \frac{2}{2} < x < \frac{7}{2} - \frac{2}{2}$$ $$-\frac{5}{2} < x < \frac{5}{2}$$ 5. **Final answer:** The solution to the inequality is $$x \in \left(-\frac{5}{2}, \frac{5}{2}\right)$$ This means $x$ is any number strictly between $-2.5$ and $2.5$.