Question

alan needs to earn a b in his geology class. his current test scores are 87, 81, 82, and 76. his final exam is worth 3 test scores. in order to earn a b alans average must lie between 80 and 89 inclusive. what range of scores can alan receive on the final exam to earn a b in the course?

$square leq$ final exam score $leq square$
(type an integer or a simplified fraction.)

Explanation:

Step1: Define variables and total scores

Let \( x \) be the score on the final exam. The final exam is worth 3 test scores, so the total number of "test scores" is \( 4 + 3 = 7 \). The sum of the current test scores is \( 87 + 81 + 82 + 76 = 326 \). The sum including the final exam (counted as 3 scores) is \( 326 + 3x \).

Step2: Set up the average inequality

The average must be between 80 and 89 inclusive. The average is \( \frac{326 + 3x}{7} \). So we set up the inequality:
\[
80 \leq \frac{326 + 3x}{7} \leq 89
\]

Step3: Solve the left inequality

Multiply all parts by 7:
\[
80 \times 7 \leq 326 + 3x \leq 89 \times 7
\]
\[
560 \leq 326 + 3x \leq 623
\]
Subtract 326 from all parts:
\[
560 - 326 \leq 3x \leq 623 - 326
\]
\[
234 \leq 3x \leq 297
\]
Divide by 3:
\[
78 \leq x \leq 99
\]

Answer:

\( 78 \leq \text{final exam score} \leq 99 \)