Context

Mathematics, with its confusing symbols, numbers, and equations, can often seem difficult and disconnected. However, these elements are tools that allow us to understand and navigate our world more effectively. For this project, we will focus on one of these elements: Inequalities. But before we dive into practice, we need to understand what inequalities are and why they are important.

Inequalities are mathematical sentences that express the relationship of inequality between two terms. They have inequality symbols such as ">", "<", ">=", and "<=". These symbols denote that one side is greater than, less than, greater than or equal to, and less than or equal to the other side, respectively. Inequalities are an essential part of mathematics and are used in various practical applications involving limits and constraints.

Now you might be thinking, "Okay, I understand, but why is this important?" Well, the concepts of inequalities are very important in a variety of fields, including sciences, engineering, economics, and even in our daily lives. For example, in economics, inequalities are used to represent production limits, budget constraints, and other aspects of financial planning. Additionally, they are also used to solve problems in sciences and engineering, such as determining the maximum and minimum limits of a chemical reaction or the behavior of a bridge when different loads are applied.

To delve deeper into the topic and prepare for the practical stages of this project, we recommend the following resources:

• Inequalities: Basic Concepts and Properties - Khan Academy
• Inequalities - Prof. Iezzi
• Inequalities in Everyday Life - Brasil Escola

Also, remember that it is important to understand the concepts before moving on to practice. Therefore, try to solve some problems and look for more examples and applications in everyday life to reinforce the importance and use of inequalities!

Practical Activity

Activity Title: The Race of Inequalities

Project Objective

• Apply the concept of inequalities in a real and fun context.
• Promote teamwork, planning, and problem-solving.
• Develop research and writing skills.

Detailed Project Description

"The Race of Inequalities" is a relay race where the members of each team (3 to 5 students) will have the mission to solve a series of challenges related to inequalities, all linked to real-world scenarios such as economics, engineering, sciences, and daily life. The project should be carried out in approximately 12 hours per student, and can be divided into several work sessions.

Required Materials

• Notebook or sheets of paper for notes and calculations.
• Computer with internet access for research.
• Mathematics textbooks and other relevant discipline books.

Step by Step

1. Research and Study: Start by studying more deeply about inequalities. Use the suggested resources in the introduction and search for more reliable sources. Try to understand how and where inequalities are applied in daily life.

2. Challenge Development: After the research, each team must develop 5 challenges related to inequalities. Each challenge should be a problem to be solved involving inequalities and be linked to a real scenario. For example, a challenge may involve calculating the maximum number of people an elevator can carry without exceeding the maximum allowed weight.

3. Challenge Resolution: The teams must then exchange challenges with each other and try to solve them. The solutions should be detailed, showing the entire thought process and calculations.

4. Report: Finally, each team must write a report containing the following topics:

   • Introduction: Describe the relevance and application of inequalities in the real world and the objective of this project.
   • Development: Explain the theoretical concepts of inequalities you used, describe in detail each challenge and its resolution, and indicate the methodology used. Present and discuss the results obtained.
   • Conclusion: Summarize the main points, present the learnings and conclusions drawn from the project.
   • Bibliography: List the sources you relied on for the project.

Project Deliverables

• 5 challenges developed by the team involving inequalities in real scenarios.
• Detailed solutions to challenges from another team.
• Report written according to the above guidelines.

Remember that the report is an important academic document that should complement the practical part of the project. In it, you should justify the solutions found and explain how you reached them, applying the theoretical concepts learned. Additionally, the report should demonstrate the group's understanding of the topic, collaboration, and time management during the project execution.