1. **Stating the problem:** We are given two conditions involving an integer $n$: - $n > 5$ - $-1 < n + 3 \leq 5$ We need to find all possible integer values of $n$ that satisfy both conditions. 2. **Analyze the first condition:** The first condition is straightforward: $$n > 5$$ This means $n$ must be an integer greater than 5, so possible values are $6, 7, 8, \ldots$. 3. **Analyze the second condition:** The second condition is: $$-1 < n + 3 \leq 5$$ We can solve this compound inequality step-by-step. First, subtract 3 from all parts: $$-1 - 3 < n + 3 - 3 \leq 5 - 3$$ which simplifies to: $$-4 < n \leq 2$$ Since $n$ is an integer, this means: $$n \in \{-3, -2, -1, 0, 1, 2\}$$ 4. **Combine both conditions:** We want integers $n$ such that: $$n > 5$$ and $$-4 < n \leq 2$$ There is no integer $n$ that is simultaneously greater than 5 and less than or equal to 2. 5. **Conclusion:** No integer values of $n$ satisfy both conditions simultaneously. **Final answer:** There are no possible values of $n$ that satisfy both $n > 5$ and $-1 < n + 3 \leq 5$.