A student scored 66 points on his Psychology 101 midterm. If the average of his midtierm 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 lests have a maximum of 100 points. Hint Wirte a compound inegrality yurth his average between 78 and 30 inclusive.

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To determine the range of final exam scores that would allow the student to receive a **B** in Psychology 101, let's follow these steps:

### 1. Understand the Given Information
- **Midterm Score (M):** 66 points
- **Final Exam Score (F):** To be determined
- **Maximum Points for Each Exam:** 100 points
- **Desired Average Range for a B:** Between 78 and 90 (inclusive)

### 2. Set Up the Average Formula
The average of the midterm and final exam scores should fall between 78 and 90. The average is calculated as:

$$
\text{Average} = \frac{M + F}{2}
$$

Substituting the known midterm score:

$$
\frac{66 + F}{2}
$$

### 3. Create the Compound Inequality
The average must be at least 78 and at most 90:

$$
78 \leq \frac{66 + F}{2} \leq 90
$$

### 4. Solve the Inequality
**Step 1:** Multiply all parts of the inequality by 2 to eliminate the denominator:

$$
2 \times 78 \leq 66 + F \leq 2 \times 90
$$

$$
156 \leq 66 + F \leq 180
$$

**Step 2:** Subtract 66 from all parts to isolate $$ F $$:

$$
156 - 66 \leq F \leq 180 - 66
$$

$$
90 \leq F \leq 114
$$

### 5. Apply the Maximum Score Constraint
Since the final exam is out of 100 points, the maximum possible score $$ F $$ can be is 100. Therefore, even though the inequality suggests $$ F \leq 114 $$, the practical upper limit is 100.

### 6. Final Range for the Final Exam Score
Combining the results:

$$
90 \leq F \leq 100
$$

**Conclusion:** The student must score **between 90 and 100 points** on the final exam to achieve an average between 78 and 90, thereby earning a **B** in the course.
To get a B, the student needs to score between 90 and 100 on the final exam.

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