If $n$ is an integer between $20$ and $80$, then any of the following could be $n + 7$ except

### Concept & Logic When establishing the possible values of an expression involving an inequality, we must substitute the boundary values of the given constraint to find the new bounds. Alternatively, we can work backwards from the options to see which one violates the initial condition. ### Step-by-Step Solution **Given:** * $n$ is an integer between $20$ and $80$. This strictly means $20 < n < 80$. * We need to identify which option falls outside the range of $n + 7$. **Calculation / Deduction:** First, establish the strict integer range for $n$: $21 \le n \le 79$ Next, find the corresponding range for $n + 7$ by adding $7$ to all parts of the inequality: $$21 + 7 \le n + 7 \le 79 + 7$$ $$28 \le n + 7 \le 86$$ This means the valid range for $n + 7$ must be anywhere between $28$ and $86$ inclusive. Let's check the options against this mathematical boundary: (a) $47$: Inside range (b) $58$: Inside range (c) $84$: Inside range (d) $88$: Outside range ($88 > 86$) ### Exam Strategy & Shortcut Instead of calculating and shifting the full inequality range, test the extreme options backwards. The largest option is $88$. If $n+7 = 88$, then $n = 81$. The question explicitly states $n$ is between $20$ and $80$. Since $81$ is greater than $80$, option (d) is immediately invalid. Reverse-checking options is highly efficient here. ### Common Pitfall Students often misinterpret "between 20 and 80" as inclusive ($20 \le n \le 80$). However, even if incorrectly interpreted inclusively, $n+7$ would max out at $80+7 = 87$. $88$ is still definitively impossible. The primary pitfall is wasting time doing forward calculation instead of using the reverse-check shortcut. ### Final Answer Therefore, the correct answer is 88.