In your​ class, you have scores of  86​, 88​, 91​, and 93 on the first four of five tests. To get a grade of Upper B​, the average of the first five tests scores must be greater than or equal to 80 and less than 90.
a.
Solve an inequality to find the least score you can get on the last test and still earn a Upper B.
b.
What score do you need if the fifth test counts as two​ tests?

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.

---

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.

Learn how to solve inequalities to determine average test scores needed for specific grades. This problem involves calculating averages and applying inequalities to find minimum score requirements for earning a Upper B grade.

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