Question

9.
A student scored 83 and 91 on their first two quizzes. Write and solve a compound inequality to find the possible values for a third quiz score that would give them an average between 85 and 90, inclusive.

Answer 1

To find the possible values for a third quiz score that would give the student an average between 85 and 90 (inclusive), we can start by finding the average of the three quiz scores. Let \( x \) be the score on the third quiz. The average of the three scores is given by:

\[
\text{Average} = \frac{83 + 91 + x}{3}
\]

We want this average to be between 85 and 90, inclusive. This can be written as a compound inequality:

\[
85 \leq \frac{83 + 91 + x}{3} \leq 90
\]

Now we can solve this compound inequality step by step.

### Step 1: Solve the left part of the inequality

Starting with:

\[
85 \leq \frac{83 + 91 + x}{3}
\]

Multiply both sides by 3 to eliminate the fraction:

\[
255 \leq 83 + 91 + x
\]

Combine the constant terms on the right:

\[
255 \leq 174 + x
\]

Now isolate \( x \):

\[
255 - 174 \leq x
\]
\[
81 \leq x
\]

This simplifies to:

\[
x \geq 81
\]

### Step 2: Solve the right part of the inequality

Now considering the right part of the compound inequality:

\[
\frac{83 + 91 + x}{3} \leq 90
\]

Multiply both sides by 3:

\[
83 + 91 + x \leq 270
\]

Again combine the constant terms:

\[
174 + x \leq 270
\]

Now isolate \( x \):

\[
x \leq 270 - 174
\]
\[
x \leq 96
\]

### Step 3: Combine the results

Now we combine both parts of the inequality:

\[
81 \leq x \leq 96
\]

### Conclusion

The possible values for the third quiz score \( x \) that would give the student an average between 85 and 90, inclusive, are:

\[
\boxed{[81, 96]}
\]