A student scored 69 points on his Psychology 101 midterm. If the average of his midterm and final must be between
78 and 90 inclusive for a B, then for what range of scores on the final exam would a student get a B? Both tests have
a maximum of 100 points. Hint: Write a compound inequality-with his average between 78 and 90 inclusive.

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UpStudy AI · Accepted Answer

To determine the range of final exam scores that would result in a **B** for the student, we'll follow these steps:

1. **Understand the Requirements:**
   - **Midterm Score:** 69 points
   - **Final Exam:** Maximum of 100 points
   - **Average Needed for a B:** Between 78 and 90 (inclusive)

2. **Set Up the Compound Inequality:**
   The average of the midterm and final exam scores should be between 78 and 90. This can be expressed as:
   $$
   78 \leq \frac{\text{Midterm} + \text{Final}}{2} \leq 90
   $$
   
3. **Plug in the Midterm Score:**
   $$
   78 \leq \frac{69 + \text{Final}}{2} \leq 90
   $$
   
4. **Multiply All Parts by 2 to Eliminate the Fraction:**
   $$
   156 \leq 69 + \text{Final} \leq 180
   $$
   
5. **Subtract 69 from All Parts to Solve for the Final Exam Score:**
   $$
   87 \leq \text{Final} \leq 111
   $$
   
6. **Consider the Maximum Possible Score:**
   Since the final exam is out of 100 points, the upper limit of 111 is not possible. Therefore, the final exam score must be **no more than 100**.

7. **Final Range:**
   $$
   87 \leq \text{Final} \leq 100
   $$

**Conclusion:**  
To achieve a **B**, the student must score **at least 87** points on the final exam and **no more than 100** points.

**Answer:**  
All final exam scores from 87 through 100 are eligible for a B.
To get a B, the student needs to score between 87 and 100 on the final exam.

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