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.