Let's break down the problem step by step.

Part (a): Solving the Inequality for the Least Score on the Last Test

To get a grade of B, the average of the first five test scores must be between 80 and 90.

Let $x$ be the score on the fifth test. The scores for the first four tests are 86, 88, 91, and 93.

The average of the five test scores is:

$\text{Average} = \frac{86 + 88 + 91 + 93 + x}{5}$

We want this average to be between 80 and 90, so:

$80 \leq \frac{86 + 88 + 91 + 93 + x}{5} < 90$

Now, let's solve this inequality.

1. Multiply all parts of the inequality by 5 to eliminate the fraction:

$80 \times 5 \leq 86 + 88 + 91 + 93 + x < 90 \times 5$

This simplifies to:

$400 \leq 358 + x < 450$

2. Subtract 358 from all parts of the inequality:

$400 - 358 \leq x < 450 - 358$

This simplifies to:

$42 \leq x < 92$

So, the least score you can get on the last test to still earn a B is 42.

Part (b): Fifth Test Counts as Two Tests

If the fifth test counts as two tests, then you have effectively six tests, where the fifth score $x$ is counted twice.

The new average would be:

$\text{Average} = \frac{86 + 88 + 91 + 93 + x + x}{6} = \frac{358 + 2x}{6}$

Again, the average must be between 80 and 90:

$80 \leq \frac{358 + 2x}{6} < 90$

Let's solve this inequality:

1. Multiply all parts of the inequality by 6:

$80 \times 6 \leq 358 + 2x < 90 \times 6$

This simplifies to:

$480 \leq 358 + 2x < 540$

2. Subtract 358 from all parts of the inequality:

$480 - 358 \leq 2x < 540 - 358$

This simplifies to:

$122 \leq 2x < 182$

3. Divide all parts by 2:

$61 \leq x < 91$

So, if the fifth test counts as two tests, you need to score at least 61 to still earn a B.

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Do you want more details on any part of the solution or have any other questions?

Here are 5 related questions:

1. What is the highest score you could have on the fifth test to still earn a B?
2. How would the required score change if the first four test scores were different?
3. What is the minimum average score to achieve an A grade?
4. How would the solution change if the fifth test counted as three tests?
5. What if there were six tests total? How would you calculate the minimum score needed?

Tip: Always check the boundaries when working with inequalities to ensure that all conditions are met.