Laura is enrolled in a one semester computer applications class. She achieves grades of $70, 86, 81$, and 83 on the first four exams. The final exam counts the same as the four exams already given.
If $x$ represents the grade on the final exam, write an expression that represents her course average.
If Laura's average is greater than or equal to 80 and less than 90, she will earn a B in the course. Write a compound inequality that must be satisfied to earn a B and solve the inequality.
a.
[tex]
\begin{array}{l}
\frac{70+86+81+83+x}{5} \\
80 \leq x\ \textless \ 130
\end{array}
[/tex]
b.
[tex]
\begin{array}{l}
\frac{70+86+81+83+x}{5} \\
80 \leq \frac{70+86+81+83+x}{5}\ \textless \ 130
\end{array}
[/tex]
c.
[tex]\frac{70+86+81+83+\pi}{5}[/tex]
Answer (2)
The expression for Laura's course average is 5 70 + 86 + 81 + 83 + x .
The compound inequality representing the condition for earning a B is 80 ≤ 5 70 + 86 + 81 + 83 + x < 90 .
Solving the inequality involves multiplying by 5 and subtracting 320, resulting in 80 ≤ x < 130 .
Laura must score between 80 and 130 (inclusive of 80, exclusive of 130) on the final exam to earn a B: 80 ≤ x < 130 .
Explanation
Understanding the Problem Let's break down this problem step by step. First, we need to determine the expression for Laura's course average. Then, we'll set up a compound inequality to find the range of scores she needs on the final exam to earn a B.
Finding the Course Average To find the course average, we sum all the grades and divide by the number of grades. Laura has four exam grades: 70, 86, 81, and 83. The final exam grade is represented by x . Since the final exam counts the same as the other exams, we have a total of 5 grades. Therefore, the expression for her course average is: 5 70 + 86 + 81 + 83 + x
Setting up the Compound Inequality Now, we need to write a compound inequality to represent the condition for Laura earning a B. A B is earned if the average is greater than or equal to 80 and less than 90. So, the compound inequality is: 80 ≤ 5 70 + 86 + 81 + 83 + x < 90
Solving the Compound Inequality Next, we solve the compound inequality. We already know that 70 + 86 + 81 + 83 = 320 . So the inequality becomes: 80 ≤ 5 320 + x < 90 To isolate x , we first multiply all parts of the inequality by 5: 80 × 5 ≤ 320 + x < 90 × 5 400 ≤ 320 + x < 450 Now, we subtract 320 from all parts of the inequality: 400 − 320 ≤ x < 450 − 320 80 ≤ x < 130
Final Answer Therefore, Laura must score between 80 and 130 (inclusive of 80, exclusive of 130) on the final exam to earn a B in the course.
Examples
In real life, calculating averages and understanding inequalities are useful in many situations. For example, if you are trying to maintain a certain grade point average (GPA) in college, you can use similar calculations to determine what grades you need to achieve in future courses. Similarly, if you are managing a budget, you can use inequalities to ensure that your expenses do not exceed your income. These mathematical tools help in planning and decision-making in various aspects of life.
Laura's course average is represented by 5 70 + 86 + 81 + 83 + x . To earn a B, her score must satisfy the compound inequality 80 ≤ x < 130 . This indicates she needs to score between 80 and 130 on her final exam.
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