A student scored 84 and 87 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 the third quiz score that would give the student an average between 85 and 90, we can set up a compound inequality.

Let \(x\) be the score of the third quiz. The average score for the three quizzes can be calculated as follows:

\[ \text{Average} = \frac{84 + 87 + x}{3} \]

We want this average to be between 85 and 90, inclusive. Therefore, we can set up the following compound inequality:

\[ 85 \leq \frac{84 + 87 + x}{3} \leq 90 \]

To eliminate the fraction, we multiply all parts of the inequality by 3:

\[ 3 \cdot 85 \leq 84 + 87 + x \leq 3 \cdot 90 \]

This simplifies to:

\[ 255 \leq 171 + x \leq 270 \]

Next, we will isolate \(x\) in both parts of the inequality.

1. For the left side: \[ 255 \leq 171 + x \implies 255 - 171 \leq x \implies 84 \leq x \]

2. For the right side: \[ 171 + x \leq 270 \implies x \leq 270 - 171 \implies x \leq 99 \]

Now we combine both results to express the compound inequality for the third quiz score \(x\):

\[ 84 \leq x \leq 99 \]

Thus, the possible values for the third quiz score that would give the student an average between 85 and 90, inclusive, are:

\[ \boxed{[84, 99]} \]