Select The Correct Answer From Each Drop-down Menu. Mary Bought More Than 72 Pencils To Distribute To Children At An Orphanage. She Packs These Pencils In Gift Boxes. If She Packs 4 Pencils Each Into X Gift Boxes, The Inequality That Represents This

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Introduction

In this problem, we are given a scenario where Mary has to distribute more than 72 pencils to children at an orphanage. She packs these pencils in gift boxes, with each box containing 4 pencils. We need to find the inequality that represents the number of gift boxes required to pack the pencils.

Understanding the Problem

Let's break down the problem step by step:

  • Mary has more than 72 pencils.
  • She packs 4 pencils in each gift box.
  • We need to find the inequality that represents the number of gift boxes required to pack the pencils.

Mathematical Representation

Let's represent the number of gift boxes as x. Since each box contains 4 pencils, the total number of pencils packed in x boxes is 4x.

We are given that Mary has more than 72 pencils, so we can write an inequality to represent this:

4x > 72

Solving the Inequality

To solve the inequality, we need to isolate the variable x. We can do this by dividing both sides of the inequality by 4:

x > 72/4

x > 18

Conclusion

Therefore, the inequality that represents the number of gift boxes required to pack the pencils is x > 18. This means that Mary needs to pack the pencils in more than 18 gift boxes.

Discussion

This problem involves mathematical modeling, where we use mathematical concepts to represent real-world situations. In this case, we used inequalities to represent the number of gift boxes required to pack the pencils. This type of problem is commonly encountered in mathematics, particularly in algebra and calculus.

Real-World Applications

Mathematical modeling has numerous real-world applications, including:

  • Business: Mathematical modeling is used to optimize business processes, such as supply chain management and resource allocation.
  • Science: Mathematical modeling is used to simulate complex systems, such as weather patterns and population dynamics.
  • Engineering: Mathematical modeling is used to design and optimize systems, such as bridges and electronic circuits.

Tips and Tricks

When solving inequalities, remember to:

  • Isolate the variable: Move all terms involving the variable to one side of the inequality.
  • Check the direction: Make sure the inequality is in the correct direction (e.g., x > 18, not x < 18).
  • Use inverse operations: Use inverse operations (e.g., addition, subtraction, multiplication, division) to isolate the variable.

Practice Problems

Try solving the following problems:

  • If a bakery sells more than 240 loaves of bread, and each loaf is packed in a box containing 6 loaves, what is the inequality that represents the number of boxes required?
  • If a company has more than 360 employees, and each employee is assigned to a team of 8 people, what is the inequality that represents the number of teams required?

Conclusion

Q&A: Mathematical Modeling for Gift Boxes

Q: What is mathematical modeling?

A: Mathematical modeling is the process of using mathematical concepts to represent real-world situations. It involves using mathematical equations, inequalities, and other mathematical tools to describe and analyze complex systems.

Q: Why is mathematical modeling important?

A: Mathematical modeling is important because it allows us to:

  • Analyze complex systems: Mathematical modeling helps us understand complex systems and make predictions about their behavior.
  • Optimize processes: Mathematical modeling helps us optimize processes and make decisions based on data.
  • Solve real-world problems: Mathematical modeling helps us solve real-world problems, such as predicting population growth or optimizing supply chains.

Q: How do I use mathematical modeling to solve inequalities?

A: To use mathematical modeling to solve inequalities, follow these steps:

  1. Identify the problem: Identify the problem you want to solve and the variables involved.
  2. Represent the problem mathematically: Represent the problem using mathematical equations or inequalities.
  3. Solve the inequality: Solve the inequality using algebraic techniques, such as isolating the variable.
  4. Check the solution: Check the solution to ensure it is valid and makes sense in the context of the problem.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

  • Linear inequalities: Inequalities that can be written in the form ax + b > c, where a, b, and c are constants.
  • Quadratic inequalities: Inequalities that can be written in the form ax^2 + bx + c > 0, where a, b, and c are constants.
  • Absolute value inequalities: Inequalities that involve absolute values, such as |x| > a, where a is a constant.

Q: How do I solve absolute value inequalities?

A: To solve absolute value inequalities, follow these steps:

  1. Write the inequality as a double inequality: Write the absolute value inequality as a double inequality, such as -a < x < a.
  2. Solve the double inequality: Solve the double inequality using algebraic techniques, such as isolating the variable.
  3. Check the solution: Check the solution to ensure it is valid and makes sense in the context of the problem.

Q: What are some real-world applications of mathematical modeling?

A: Some real-world applications of mathematical modeling include:

  • Business: Mathematical modeling is used to optimize business processes, such as supply chain management and resource allocation.
  • Science: Mathematical modeling is used to simulate complex systems, such as weather patterns and population dynamics.
  • Engineering: Mathematical modeling is used to design and optimize systems, such as bridges and electronic circuits.

Q: How can I practice mathematical modeling?

A: To practice mathematical modeling, try the following:

  • Work on practice problems: Work on practice problems, such as those found in math textbooks or online resources.
  • Use real-world examples: Use real-world examples to illustrate mathematical concepts and apply them to practical problems.
  • Join a math club or study group: Join a math club or study group to collaborate with others and learn from their experiences.

Conclusion

In conclusion, mathematical modeling is a powerful tool for representing real-world situations using mathematical concepts. By following the steps outlined in this article, you can develop your skills in mathematical modeling and apply them to a wide range of real-world problems. Remember to practice mathematical modeling regularly to improve your skills and stay up-to-date with the latest developments in the field.