Question

discussion 2: target exam score
amber has scores 75.7, 68.2, and 83.9 on the three midterm tests of her biology class.
a. write down an inequality for finding the score that she must make on the final exam to pass the course with an average of 80 or higher, given that the final exam counts as two tests. make sure the inequality that your type has proper grouping symbols.
b. after solving the inequality, explain in a complete sentence the meaning of the solution.
note:

1. required: typeset the inequality using the insert math equation button (icon √x). pictures of handwritten response are not accepted. this will also be a requirement for all future discussions.
2. please carefully proofread your work before hitting the post reply button because the discussion board setting does not allow your response to be deleted or edited once posted.

Explanation:

Part (a)

Step1: Define the variable

Let \( x \) be the score on the final exam. Since the final exam counts as two tests, the total number of test scores considered is \( 3 + 2=5 \) (3 midterms and 2 equivalent from the final). The sum of the midterm scores is \( 75.7 + 68.2+83.9 \), and the sum of all scores (including the final counted twice) is \( 75.7 + 68.2 + 83.9+2x \). The average is the sum divided by the number of tests, and we want the average to be at least 80.

Step2: Set up the inequality

The average is \( \frac{75.7 + 68.2+83.9 + 2x}{5} \), and we want this to be greater than or equal to 80. So the inequality is:
\[
\frac{75.7 + 68.2+83.9 + 2x}{5}\geq80
\]

Part (b)

Step1: Solve the inequality

First, simplify the numerator: \( 75.7+68.2 + 83.9=227.8 \). So the inequality becomes \( \frac{227.8 + 2x}{5}\geq80 \). Multiply both sides by 5: \( 227.8+2x\geq400 \). Subtract 227.8 from both sides: \( 2x\geq400 - 227.8=172.2 \). Divide both sides by 2: \( x\geq86.1 \).

Step2: Interpret the solution

The solution \( x\geq86.1 \) means that Amber must score 86.1 or higher on the final exam (where the final exam is weighted as two test scores) in order to have a course average of 80 or higher.

Answer:

(Part a):
\[
\boldsymbol{\frac{75.7 + 68.2+83.9 + 2x}{5}\geq80}
\]