1. **State the problem:** We are given integers $x$ and $y$ such that $3 < x < 7$, $4 < y < 9$, and $x + y = 13$. We need to find all possible values of $x$. 2. **Analyze the inequalities:** Since $x$ and $y$ are integers, - $3 < x < 7$ means $x$ can be $4, 5,$ or $6$. - $4 < y < 9$ means $y$ can be $5, 6, 7,$ or $8$. 3. **Use the equation $x + y = 13$:** For each possible $x$, find $y = 13 - x$ and check if $y$ satisfies $4 < y < 9$. - For $x=4$, $y=13-4=9$ (not less than 9, so invalid). - For $x=5$, $y=13-5=8$ (valid since $5 < 8 < 9$). - For $x=6$, $y=13-6=7$ (valid since $4 < 7 < 9$). 4. **Conclusion:** The possible values of $x$ are $5$ and $6$. **Final answer:** $\boxed{5, 6}$