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 you type has proper grouping symbols.
b. after solving the inequality, explain in a complete sentence the meaning of the solution.
note:

1. required: type - set 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 variables and total tests

Let \( x \) be the score on the final exam. The final exam counts as two tests, so the total number of "test scores" is \( 3 + 2 = 5 \) (3 midterms + 2 final equivalents). The sum of the midterm scores is \( 75.7 + 68.2 + 83.9 \), and the sum of the final exam scores (counted twice) is \( 2x \). The average is the total sum divided by the total number of tests, and we want this average to be at least 80.
\[
\text{Average} = \frac{75.7 + 68.2 + 83.9 + 2x}{5}
\]

Step2: Set up the inequality

We want the average to be 80 or higher, so:
\[
\frac{75.7 + 68.2 + 83.9 + 2x}{5} \geq 80
\]
First, calculate the sum of the midterms: \( 75.7 + 68.2 + 83.9 = 227.8 \). So the inequality becomes:
\[
\frac{227.8 + 2x}{5} \geq 80
\]

Part (b)

Step1: Solve the inequality

Multiply both sides by 5: \( 227.8 + 2x \geq 400 \)
Subtract 227.8 from both sides: \( 2x \geq 400 - 227.8 = 172.2 \)
Divide both sides by 2: \( x \geq \frac{172.2}{2} = 86.1 \)

Step2: Interpret the solution

The solution \( x \geq 86.1 \) means that Amber must score 86.1 or higher on the final exam (where the final exam is weighted as two test scores) to have an average of 80 or higher in the biology class.

Part (a) Answer:

\(\boldsymbol{\frac{75.7 + 68.2 + 83.9 + 2x}{5} \geq 80}\) (or simplified as \(\boldsymbol{\frac{227.8 + 2x}{5} \geq 80}\))

Part (b) Answer:

Amber must score 86.1 or higher on the final exam (which counts as two test scores) to achieve an average of 80 or higher in her biology class.

Answer:

Step1: Solve the inequality

Multiply both sides by 5: \( 227.8 + 2x \geq 400 \)
Subtract 227.8 from both sides: \( 2x \geq 400 - 227.8 = 172.2 \)
Divide both sides by 2: \( x \geq \frac{172.2}{2} = 86.1 \)

Step2: Interpret the solution

The solution \( x \geq 86.1 \) means that Amber must score 86.1 or higher on the final exam (where the final exam is weighted as two test scores) to have an average of 80 or higher in the biology class.

Part (a) Answer:

\(\boldsymbol{\frac{75.7 + 68.2 + 83.9 + 2x}{5} \geq 80}\) (or simplified as \(\boldsymbol{\frac{227.8 + 2x}{5} \geq 80}\))

Part (b) Answer:

Amber must score 86.1 or higher on the final exam (which counts as two test scores) to achieve an average of 80 or higher in her biology class.