1. **Problem statement:** Peter scored 78 on his mid-semester exam. To get grade A, the average of his mid-semester and final exam scores must be between 80 and 88 inclusive. 2. **Formula:** The average score is given by $$\text{Average} = \frac{\text{Mid-semester score} + \text{Final exam score}}{2}$$ 3. **Set up inequalities:** We want $$80 \leq \frac{78 + x}{2} \leq 88$$ where $x$ is the final exam score. 4. **Solve the inequalities:** Multiply all parts by 2: $$160 \leq 78 + x \leq 176$$ Subtract 78 from all parts: $$160 - 78 \leq x \leq 176 - 78$$ Simplify: $$82 \leq x \leq 98$$ 5. **Interpretation:** Peter's final exam score must be at least 82 and at most 98 to get grade A. **Final answer:** The final exam score must lie in the range $$[82, 98]$$ inclusive.