In a math class, a student has scores of 82, 85, 98, and 95 on the first four exams. Assuming that 100 is the maximum number of points on each test, what must the range of scores be on the fifth exam so that the average of the five tests is greater than or equal to 90? Answer in interval notation.

To find the range of scores on the fifth exam that would ensure the average of all five exams is greater than or equal to 90, we need to follow these steps:

### Step 1: Calculate the total score needed for an average of 90
The average score of the five exams must be greater than or equal to 90. If \( x \) is the score on the fifth exam, the average of all five exams is:

\[
\text{Average} = \frac{82 + 85 + 98 + 95 + x}{5}
\]

We want this average to be at least 90:

\[
\frac{82 + 85 + 98 + 95 + x}{5} \geq 90
\]

### Step 2: Solve the inequality
First, calculate the sum of the scores on the first four exams:

\[
82 + 85 + 98 + 95 = 360
\]

Substitute this into the inequality:

\[
\frac{360 + x}{5} \geq 90
\]

Multiply both sides by 5 to eliminate the fraction:

\[
360 + x \geq 450
\]

Now, solve for \( x \):

\[
x \geq 450 - 360
\]

\[
x \geq 90
\]

### Step 3: Determine the interval
Since the maximum score on any exam is 100, \( x \) must satisfy:

\[
90 \leq x \leq 100
\]

### Final Answer: Interval Notation
The range of scores on the fifth exam that would give the student an average of at least 90 is:

\[
\boxed{[90, 100]}
\]

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions:
1. How would the interval change if the required average was 95?
2. What is the lowest possible average if the fifth score is 70?
3. How many points above 90 is the student's current average without the fifth score?
4. What is the maximum possible average if the fifth score is 100?
5. If the score on the fifth exam was 85, what would the average of the five tests be?

**Tip:** Always double-check your calculations when dealing with inequalities, especially when determining interval ranges.

Learn how to calculate the range of scores needed on the fifth exam to achieve an average of at least 90 across five exams. This problem involves arithmetic calculations, inequalities, and average calculations. Suitable for Grades 9-10.

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