Question

1. cindy has scores of 72, 82, 83, and 89 on her biology tests. use a compound inequality to find the range of scores she can make on her final exam to receive a c in the course. the final exam counts as two tests, and a c is received if the final course average is from 77 to 84.

Explanation:

Step1: Calculate the sum of existing scores

The sum of Cindy's scores on the four biology tests is \(72 + 82+83 + 89=326\).

Step2: Set up the compound - inequality for the average

Let \(x\) be the score on the final exam. Since the final exam counts as two tests, the total number of "test - equivalents" is \(4 + 2=6\).
The average \(\bar{x}=\frac{326 + 2x}{6}\).
We know that \(77\leqslant\frac{326+2x}{6}\leqslant84\).

Step3: Solve the left - hand side of the inequality

Multiply each part of the compound inequality by \(6\): \(77\times6\leqslant326 + 2x\).
\(462\leqslant326+2x\).
Subtract \(326\) from both sides: \(462-326\leqslant2x\), \(136\leqslant2x\).
Divide by \(2\): \(x\geqslant68\).

Step4: Solve the right - hand side of the inequality

We also have \(326 + 2x\leqslant84\times6\).
\(326+2x\leqslant504\).
Subtract \(326\) from both sides: \(2x\leqslant504 - 326\), \(2x\leqslant178\).
Divide by \(2\): \(x\leqslant89\).

Answer:

Cindy's score \(x\) on the final exam must satisfy the compound inequality \(68\leqslant x\leqslant89\)