Representing the System of Inequalities

Oct 13, 2025 by leolavanderia.com.br 33 views

Introduction

In this article, we will delve into the world of mathematics, specifically systems of inequalities, to solve a real-world problem. Zahra, a crafty individual, is preparing for a craft fair by knitting items to sell. She has a total of 2,640 yards of yarn and wants to create a minimum of 15 items. The question is, how many scarves and hats can she make with the yarn she has? This problem can be represented using a system of inequalities, which we will explore in this article.

Understanding the Problem

Zahra has 2,640 yards of yarn, and she wants to make a minimum of 15 items. Each scarf uses 200 yards of yarn, and each hat uses 150 yards. We can represent the number of scarves and hats she can make using the following variables:

S: number of scarves
H: number of hats

We know that the total number of yards of yarn used is equal to the sum of the yards used for each scarf and each hat. This can be represented by the equation:

200S + 150H ≤ 2640

We also know that Zahra wants to make a minimum of 15 items, so the total number of scarves and hats she makes must be greater than or equal to 15. This can be represented by the inequality:

S + H ≥ 15

Representing the System of Inequalities

The system of inequalities representing Zahra's yarn conundrum is:

200S + 150H ≤ 2640 S + H ≥ 15

Solving the System of Inequalities

To solve this system of inequalities, we can use a graphical approach. We will graph the two inequalities on a coordinate plane and find the region where they intersect.

Graphing the First Inequality

The first inequality is 200S + 150H ≤ 2640. We can rewrite this inequality as:

H ≤ (-200/150)S + 17.6

This is a linear inequality in the form of H ≤ mx + b, where m is the slope and b is the y-intercept. We can graph this inequality by plotting the line H = (-200/150)S + 17.6 and shading the region below it.

Graphing the Second Inequality

The second inequality is S + H ≥ 15. We can rewrite this inequality as:

H ≥ -S + 15

This is also a linear inequality in the form of H ≥ mx + b. We can graph this inequality by plotting the line H = -S + 15 and shading the region above it.

Finding the Intersection Region

The region where the two inequalities intersect is the solution to the system of inequalities. We can find this region by looking for the area where the two shaded regions overlap.

Analyzing the Solution

The solution to the system of inequalities is the region where Zahra can make a minimum of 15 items with the yarn she has. We can analyze this solution by looking at the boundaries of the region.

The boundary of the first inequality is the line H = (-200/150)S + 17.6. This line represents the maximum number of hats Zahra can make for each scarf.
The boundary of the second inequality is the line H = -S + 15. This line represents the minimum number of hats Zahra must make for each scarf.

Conclusion

In this article, we have explored the problem of Zahra's yarn conundrum using a system of inequalities. We have represented the problem using two linear inequalities and solved the system using a graphical approach. The solution to the system of inequalities is the region where Zahra can make a minimum of 15 items with the yarn she has. We have analyzed this solution by looking at the boundaries of the region and have found that Zahra can make a maximum of 13 scarves and 2 hats, or a maximum of 12 scarves and 3 hats, or a maximum of 11 scarves and 4 hats, and so on.

Final Answer

The final answer to the problem is that Zahra can make a minimum of 15 items with the yarn she has. The solution to the system of inequalities is the region where the two shaded regions overlap, and the boundaries of this region represent the maximum number of scarves and hats Zahra can make.

References

[1] "Systems of Inequalities" by Math Open Reference
[2] "Graphing Linear Inequalities" by Khan Academy
[3] "Solving Systems of Inequalities" by Purplemath

Additional Resources

[1] "Mathematics for Crafters" by Craftsy
[2] "Yarn and Fiber Arts" by The Spruce Crafts
[3] "Math in Real Life" by Math Is Fun
Q&A: Zahra's Yarn Conundrum =============================

Introduction

In our previous article, we explored the problem of Zahra's yarn conundrum using a system of inequalities. We represented the problem using two linear inequalities and solved the system using a graphical approach. In this article, we will answer some frequently asked questions about Zahra's yarn conundrum.

Q: What is the main goal of Zahra's yarn conundrum?

A: The main goal of Zahra's yarn conundrum is to determine how many scarves and hats she can make with the yarn she has, while making a minimum of 15 items.

Q: What are the two inequalities that represent Zahra's yarn conundrum?

A: The two inequalities that represent Zahra's yarn conundrum are:

200S + 150H ≤ 2640 S + H ≥ 15

Q: How can we solve the system of inequalities?

A: We can solve the system of inequalities using a graphical approach. We can graph the two inequalities on a coordinate plane and find the region where they intersect.

Q: What is the solution to the system of inequalities?

A: The solution to the system of inequalities is the region where Zahra can make a minimum of 15 items with the yarn she has. We can analyze this solution by looking at the boundaries of the region.

Q: What are the boundaries of the solution region?

A: The boundaries of the solution region are the lines H = (-200/150)S + 17.6 and H = -S + 15. These lines represent the maximum number of hats Zahra can make for each scarf and the minimum number of hats Zahra must make for each scarf.

Q: What are some possible solutions to the system of inequalities?

A: Some possible solutions to the system of inequalities are:

13 scarves and 2 hats
12 scarves and 3 hats
11 scarves and 4 hats
10 scarves and 5 hats

Q: How can we determine the number of scarves and hats Zahra can make?

A: We can determine the number of scarves and hats Zahra can make by looking at the boundaries of the solution region. We can use the lines H = (-200/150)S + 17.6 and H = -S + 15 to find the maximum number of hats Zahra can make for each scarf and the minimum number of hats Zahra must make for each scarf.

Q: What are some real-world applications of systems of inequalities?

A: Some real-world applications of systems of inequalities include:

Resource allocation: Systems of inequalities can be used to allocate resources such as yarn, fabric, or other materials.
Budgeting: Systems of inequalities can be used to create a budget and determine how much money can be spent on different items.
Scheduling: Systems of inequalities can be used to schedule tasks and determine how much time can be spent on different activities.

Conclusion

In this article, we have answered some frequently asked questions about Zahra's yarn conundrum. We have explored the problem using a system of inequalities and solved the system using a graphical approach. We have also discussed some real-world applications of systems of inequalities and provided some possible solutions to the system of inequalities.