Problem 42 - Algebra 2

Question

REVIEW To be a member of the marching band, a student must have a GPA of at least 2.0 and must have attended at least five after-school practices. Choose the system of inequalities that best represents this situation. $$ \begin{array}{ll}{\mathbf{F} \quad x \geq 2} & {\mathbf{H} x<2} \\ {y \geq 5} & {y<5} \\ {\mathbf{G} x \leq 2} & {\mathbf{J} \quad x>2} \\ {y \leq 5} & {y>5}\end{array} $$

Step-by-Step Solution

Answer

Option **F** with \( x \geq 2 \) and \( y \geq 5 \) is the correct system.

Step 1: Identify the Variables

In this context, we need two variables to represent the conditions. Let \( x \) represent the GPA of a student and \( y \) represent the number of after-school practices attended by the student.

Step 2: Understand the Conditions

According to the problem, a student must have a GPA of at least 2.0, meaning \( x \geq 2 \). The student must also have attended at least five practices, implying \( y \geq 5 \).

Step 3: Match the Conditions to Inequalities

Now, we need to match the conditions to inequalities from the given options:- For GPA: The condition \( x \geq 2 \) matches with inequality \( \mathbf{F} \, x \geq 2 \).- For practices: The condition \( y \geq 5 \) matches with inequality \( y \geq 5 \).

Step 4: Select the Correct System of Inequalities

The correct system of inequalities representing both conditions is:- \( x \geq 2 \) (Option \( F \))- \( y \geq 5 \)Thus, the correct answer is the pair that contains both \( x \geq 2 \) and \( y \geq 5 \).

Key Concepts

GPA requirement

Grade Point Average (GPA) is a key metric used to assess a student's academic performance. In this exercise, students must have a minimum GPA of 2.0 to qualify for the marching band.

What does "at least 2.0" mean exactly? It means that the student's GPA can be 2.0 or higher. If a student's GPA is below 2.0, they are not eligible for band membership.

The inequalities representing this are expressed as:

• \(x \geq 2\)

Here, \(x\) stands for the student's GPA. This inequality ensures that only those who meet the academic requirement can join the band.

after-school practice

Involvement in extracurricular activities often requires dedication beyond school hours. Here, students must attend at least five after-school practices to be eligible for the marching band.

This ensures that students are committed and adequately prepared for performances. The term "at least five" signifies that the minimum number of practices attended should be five or more.

In terms of inequalities, this can be expressed as:

• \(y \geq 5\)

Where \(y\) represents the number of practices attended. Meeting or exceeding this threshold shows the student's commitment to the group.

inequalities

Inequalities are mathematical expressions used to represent a range of values, typically in relation to resolving conditions in situations like membership criteria. In this exercise:

• The GPA requirement can be described by the inequality \(x \geq 2\).
• The after-school practice requirement is \(y \geq 5\).

These inequalities ensure that all students meet both academic and participatory standards to qualify.

Using inequalities is an effective way to succinctly state conditions that students must satisfy, and they play a crucial role in determining eligibility by setting quantifiable limits.

student eligibility

Eligibility criteria for activities like the marching band ensure that students maintain a balance between academics and extracurricular commitments. In simpler terms, only students who meet both the GPA and the practice requirements can join.

To determine if a student is eligible, both conditions need to be true:

• The student's GPA should be 2.0 or greater (\(x \geq 2\)).
• The student must have attended at least five practices (\(y \geq 5\)).

When both these inequalities are satisfied, a student is considered eligible.

This structured approach helps maintain a standard within the activity and promotes balanced development for the student.