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