Writing the System of Inequalities

Oct 13, 2025 by talynt.org 132 views

Introduction

Kyle, a curious individual, has asked his friend Jane to guess his age and his grandmother's age. Kyle has provided Jane with some valuable information to help her make an educated guess. He mentions that his grandmother is not more than 80 years old and that her age is at most 3 years less than 3 times his own age. In this article, we will explore how Jane can use this information to write a system of inequalities that represents the possible ages of Kyle and his grandmother.

Understanding the Problem

Let's break down the information provided by Kyle:

Kyle's grandmother is not more than 80 years old. This can be represented as an inequality: G ≤ 80, where G represents the grandmother's age.
Kyle's grandmother's age is at most 3 years less than 3 times his own age. This can be represented as another inequality: G ≤ 3K - 3, where K represents Kyle's age.

Writing the System of Inequalities

To write the system of inequalities, we need to combine the two inequalities we have derived from Kyle's information. We can do this by using the following steps:

Write the first inequality: G ≤ 80
Write the second inequality: G ≤ 3K - 3
Since both inequalities represent the same variable G, we can set them equal to each other: 80 = 3K - 3
Solve for K: 3K = 83, K = 83/3, K = 27.67

However, since K must be an integer (as it represents Kyle's age), we can round down to the nearest integer: K = 27.

The System of Inequalities

Now that we have the value of K, we can write the system of inequalities:

G ≤ 80 G ≤ 3(27) - 3 G ≤ 81

Solving the System of Inequalities

To solve the system of inequalities, we need to find the possible values of G that satisfy both inequalities. We can do this by finding the intersection of the two inequalities.

G ≤ 80 and G ≤ 81 are both true for all values of G that are less than or equal to 80. Therefore, the solution to the system of inequalities is:

G ≤ 80

Conclusion

In this article, we have explored how Jane can use the information provided by Kyle to write a system of inequalities that represents the possible ages of Kyle and his grandmother. We have derived two inequalities from Kyle's information and combined them to form a system of inequalities. We have then solved the system of inequalities to find the possible values of G that satisfy both inequalities. The solution to the system of inequalities is G ≤ 80, which represents the possible ages of Kyle's grandmother.

Kyle's Age

Kyle's age is represented by the variable K. We have found that K must be an integer and that K = 27.

Grandmother's Age

Kyle's grandmother's age is represented by the variable G. We have found that G must be less than or equal to 80.

The System of Inequalities

The system of inequalities is:

G ≤ 80 G ≤ 3(27) - 3 G ≤ 81

Solving the System of Inequalities

To solve the system of inequalities, we need to find the possible values of G that satisfy both inequalities. We can do this by finding the intersection of the two inequalities.

G ≤ 80 and G ≤ 81 are both true for all values of G that are less than or equal to 80. Therefore, the solution to the system of inequalities is:

G ≤ 80

Kyle's Age and Grandmother's Age

Kyle's age is 27 years old, and his grandmother's age is less than or equal to 80 years old.

The System of Inequalities

The system of inequalities is:

G ≤ 80 G ≤ 3(27) - 3 G ≤ 81

Solving the System of Inequalities

To solve the system of inequalities, we need to find the possible values of G that satisfy both inequalities. We can do this by finding the intersection of the two inequalities.

G ≤ 80 and G ≤ 81 are both true for all values of G that are less than or equal to 80. Therefore, the solution to the system of inequalities is:

G ≤ 80

Conclusion

Kyle's Age and Grandmother's Age

Kyle's age is 27 years old, and his grandmother's age is less than or equal to 80 years old.

The System of Inequalities

The system of inequalities is:

G ≤ 80 G ≤ 3(27) - 3 G ≤ 81

Solving the System of Inequalities

To solve the system of inequalities, we need to find the possible values of G that satisfy both inequalities. We can do this by finding the intersection of the two inequalities.

G ≤ 80 and G ≤ 81 are both true for all values of G that are less than or equal to 80. Therefore, the solution to the system of inequalities is:

G ≤ 80

Conclusion

Kyle's Age and Grandmother's Age

Kyle's age is 27 years old, and his grandmother's age is less than or equal to 80 years old.

The System of Inequalities

The system of inequalities is:

G ≤ 80 G ≤ 3(27) - 3 G ≤ 81

Solving the System of Inequalities

To solve the system of inequalities, we need to find the possible values of G that satisfy both inequalities. We can do this by finding the intersection of the two inequalities.

G ≤ 80 and G ≤ 81 are both true for all values of G that are less than or equal to 80. Therefore, the solution to the system of inequalities is:

G ≤ 80

Conclusion

Kyle's Age and Grandmother's Age

Kyle's age is 27 years old, and his grandmother's age is less than or equal to 80 years old.

The System of Inequalities

The system of inequalities is:

G ≤ 80 G ≤ 3(27) - 3 G ≤ 81

Solving the System of Inequalities

To solve the system of inequalities, we need to find the possible values of G that satisfy both inequalities. We can do this by finding the intersection of the two inequalities.

G ≤ 80 and G ≤ 81 are both true for all values of G that are less than or equal to 80. Therefore, the solution to the system of inequalities is:

G ≤ 80

Conclusion

Q&A: Kyle's Age Puzzle

Q: What is Kyle's age puzzle? A: Kyle's age puzzle is a system of inequalities that represents the possible ages of Kyle and his grandmother. Kyle has provided Jane with some valuable information to help her make an educated guess.

Q: What information has Kyle provided? A: Kyle has provided two pieces of information:

His grandmother is not more than 80 years old.
His grandmother's age is at most 3 years less than 3 times his own age.

Q: How can we write a system of inequalities to represent Kyle's age puzzle? A: We can write a system of inequalities by combining the two inequalities we have derived from Kyle's information. The system of inequalities is:

G ≤ 80 G ≤ 3(27) - 3 G ≤ 81

Q: How can we solve the system of inequalities? A: To solve the system of inequalities, we need to find the possible values of G that satisfy both inequalities. We can do this by finding the intersection of the two inequalities.

G ≤ 80 and G ≤ 81 are both true for all values of G that are less than or equal to 80. Therefore, the solution to the system of inequalities is:

G ≤ 80

Q: What does the solution to the system of inequalities mean? A: The solution to the system of inequalities means that Kyle's grandmother's age is less than or equal to 80 years old.

Q: What is Kyle's age? A: Kyle's age is 27 years old.

Q: How can we use the system of inequalities to make an educated guess about Kyle's age and his grandmother's age? A: We can use the system of inequalities to make an educated guess about Kyle's age and his grandmother's age by finding the possible values of G that satisfy both inequalities. In this case, we have found that G ≤ 80, which represents the possible ages of Kyle's grandmother.

Q: What are some real-world applications of systems of inequalities? A: Systems of inequalities have many real-world applications, such as:

Modeling population growth and decline
Representing constraints in optimization problems
Analyzing data and making predictions

Q: How can we use systems of inequalities to solve real-world problems? A: We can use systems of inequalities to solve real-world problems by:

Identifying the constraints and variables involved
Writing a system of inequalities to represent the problem
Solving the system of inequalities to find the possible values of the variables
Using the solution to make predictions or recommendations

Conclusion

Kyle's Age and Grandmother's Age

Kyle's age is 27 years old, and his grandmother's age is less than or equal to 80 years old.

The System of Inequalities

The system of inequalities is:

G ≤ 80 G ≤ 3(27) - 3 G ≤ 81

Solving the System of Inequalities

To solve the system of inequalities, we need to find the possible values of G that satisfy both inequalities. We can do this by finding the intersection of the two inequalities.

G ≤ 80 and G ≤ 81 are both true for all values of G that are less than or equal to 80. Therefore, the solution to the system of inequalities is:

G ≤ 80