Representing the Problem as a System of Inequalities

Oct 13, 2025 by vinooalluri.com 81 views

Introduction

In this article, we will explore the problem of printing copies within a time limit. Julissa needs to print no fewer than 8 copies within 25 minutes. The time it takes to print a color copy is 4 minutes, and the time it takes to print a grayscale copy is 2 minutes. We will represent this situation using a system of inequalities.

Understanding the Problem

Let's break down the problem:

Julissa needs to print no fewer than 8 copies.
The time it takes to print a color copy is 4 minutes.
The time it takes to print a grayscale copy is 2 minutes.
Julissa has 25 minutes to complete the task.

We can represent the number of color copies as c and the number of grayscale copies as g. The total number of copies is c + g, and the total time taken is 4c + 2g.

Representing the Problem as a System of Inequalities

We can represent the problem as a system of inequalities:

c + g ≥ 8 (Julissa needs to print no fewer than 8 copies)
4c + 2g ≤ 25 (Julissa has 25 minutes to complete the task)

We can simplify the second inequality by dividing both sides by 2:

2c + g ≤ 12.5

However, since we cannot have a fraction of a copy, we can round down to the nearest whole number:

2c + g ≤ 12

Solving the System of Inequalities

To solve the system of inequalities, we can graph the two inequalities on a coordinate plane.

The first inequality, c + g ≥ 8, represents a line with a slope of -1 and a y-intercept of 8. The second inequality, 2c + g ≤ 12, represents a line with a slope of -2 and a y-intercept of 12.

We can graph these lines and shade the region that satisfies both inequalities.

Graphing the Inequalities

To graph the first inequality, c + g ≥ 8, we can plot the line c + g = 8 and shade the region above the line.

To graph the second inequality, 2c + g ≤ 12, we can plot the line 2c + g = 12 and shade the region below the line.

The region that satisfies both inequalities is the area where the two shaded regions overlap.

Conclusion

In this article, we represented the problem of printing copies within a time limit using a system of inequalities. We graphed the two inequalities on a coordinate plane and shaded the region that satisfies both inequalities. This region represents the possible values of c and g that satisfy the conditions of the problem.

Final Answer

The system of inequalities that represents the problem is:

c + g ≥ 8
2c + g ≤ 12

This system of inequalities represents the possible values of c and g that satisfy the conditions of the problem.

Additional Information

The time it takes to print a color copy is 4 minutes.
The time it takes to print a grayscale copy is 2 minutes.
Julissa has 25 minutes to complete the task.

References

[1] "Systems of Inequalities" by Math Open Reference
[2] "Graphing Inequalities" by Khan Academy
Julissa's Printing Problem: A Q&A Guide =============================================

Introduction

In our previous article, we explored the problem of printing copies within a time limit. Julissa needs to print no fewer than 8 copies within 25 minutes. The time it takes to print a color copy is 4 minutes, and the time it takes to print a grayscale copy is 2 minutes. We represented this situation using a system of inequalities. In this article, we will answer some frequently asked questions about Julissa's printing problem.

Q&A

Q: What is the minimum number of copies that Julissa needs to print?

A: Julissa needs to print no fewer than 8 copies.

Q: How long does it take to print a color copy?

A: It takes 4 minutes to print a color copy.

Q: How long does it take to print a grayscale copy?

A: It takes 2 minutes to print a grayscale copy.

Q: How much time does Julissa have to complete the task?

A: Julissa has 25 minutes to complete the task.

Q: What is the system of inequalities that represents the problem?

A: The system of inequalities is:

c + g ≥ 8
2c + g ≤ 12

Q: How can I graph the system of inequalities?

A: You can graph the two inequalities on a coordinate plane. Plot the line c + g = 8 and shade the region above the line. Plot the line 2c + g = 12 and shade the region below the line. The region that satisfies both inequalities is the area where the two shaded regions overlap.

Q: What is the significance of the system of inequalities?

A: The system of inequalities represents the possible values of c and g that satisfy the conditions of the problem. It helps us understand the relationship between the number of color copies and grayscale copies that Julissa needs to print.

Q: Can I use the system of inequalities to find the maximum number of copies that Julissa can print?

A: Yes, you can use the system of inequalities to find the maximum number of copies that Julissa can print. To do this, you can graph the two inequalities and find the point where the two lines intersect. This point represents the maximum number of copies that Julissa can print.

Q: What is the maximum number of copies that Julissa can print?

A: To find the maximum number of copies that Julissa can print, you can graph the two inequalities and find the point where the two lines intersect. This point represents the maximum number of copies that Julissa can print.

Conclusion

In this article, we answered some frequently asked questions about Julissa's printing problem. We provided a system of inequalities that represents the problem and explained how to graph the system of inequalities. We also discussed the significance of the system of inequalities and how it can be used to find the maximum number of copies that Julissa can print.

Additional Information

The time it takes to print a color copy is 4 minutes.
The time it takes to print a grayscale copy is 2 minutes.
Julissa has 25 minutes to complete the task.

References

[1] "Systems of Inequalities" by Math Open Reference
[2] "Graphing Inequalities" by Khan Academy