in a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Lisa has scored 94, 85, and 78
on the first three. What range of scores on the fourth test will give Lisa a B for the semester (an average between 80 and 89, inclusive)? Assume th
all test scores have a non-negative value
Express your answer in interval notation.

To find the range of scores Lisa needs on her fourth test to achieve a B for the semester, we can use the following approach:

### Step 1: Define the average requirement
Lisa needs a **B**, which means her average score must be between 80 and 89, inclusive. Therefore, we want:
\[
80 \leq \text{average} \leq 89
\]

### Step 2: Set up the equation for the average
The average is the sum of the four test scores divided by 4. Let \( x \) represent the score on the fourth test. Lisa's first three test scores are 94, 85, and 78, so her average is:
\[
\text{average} = \frac{94 + 85 + 78 + x}{4}
\]

### Step 3: Solve the inequalities
We now substitute this expression for the average into the inequality for a B grade:

\[
80 \leq \frac{94 + 85 + 78 + x}{4} \leq 89
\]

Simplify the sum of the known scores:
\[
94 + 85 + 78 = 257
\]

So the inequality becomes:
\[
80 \leq \frac{257 + x}{4} \leq 89
\]

### Step 4: Solve for \( x \)

#### Inequality 1: Solve \( 80 \leq \frac{257 + x}{4} \)
Multiply both sides by 4:
\[
320 \leq 257 + x
\]
Subtract 257 from both sides:
\[
x \geq 63
\]

#### Inequality 2: Solve \( \frac{257 + x}{4} \leq 89 \)
Multiply both sides by 4:
\[
257 + x \leq 356
\]
Subtract 257 from both sides:
\[
x \leq 99
\]

### Step 5: Final answer in interval notation
The range of scores Lisa needs on her fourth test to get a B for the semester is:
\[
x \in [63, 99]
\]

### Summary
Lisa must score between **63 and 99** on her fourth test to achieve a B for the semester.

Would you like further clarification on any part of this process?

---

Here are 5 additional questions related to this problem:

1. How would the solution change if Lisa needed an A (an average of 90 or higher)?
2. What happens if the tests are not equally weighted?
3. If Lisa wanted to achieve an exact average of 85, what score would she need on her fourth test?
4. How does the range of scores change if Lisa's lowest test score is dropped from the average calculation?
5. What would the interval be if Lisa had already taken four tests and wanted to calculate her overall grade with an additional extra-credit assignment?

**Tip**: When solving inequalities with averages, make sure to carefully handle the boundaries of the range (e.g., inclusivity of endpoints).

In a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Lisa has scored 94, 85, and 78 on the first three. What range of scores on the fourth test will give Lisa a B for the semester (an average between 80 and 89, inclusive)? Assume all test scores have a non-negative value. Express your answer in interval notation.

Determine the score Lisa needs on her fourth test to achieve a B for the semester. This problem involves calculating averages and solving inequalities. Suitable for students in Grades 7-10.

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