Context

Mathematics is a foundational tool that we encounter in our day-to-day lives and it enables us to understand and interact with the world more effectively. Of the various branches of mathematics, we have Algebra, and within it, we find inequalities.

Inequalities are mathematical expressions that, instead of equality (as in equations), express the idea of inequality. They are, therefore, an important tool for solving problems that involve limits, greater or lesser quantities, capacity, among others. In other words, it's as if we had a scale in which the weights on each side are different.

In practical life, inequalities are seen in various situations such as, for example, in economics to define minimum and maximum prices, in situations of food restriction where it's necessary to consume a greater or lesser quantity of certain nutrients, and even in engineering to calculate the maximum resistance of materials.

To delve deeper into this topic, we suggest the following reliable sources:

1. Book: Fundamentos de Matemática Elementar, vol. 1: Conjuntos, Funções, Noções de Logaritmos, Complexos (Gelson Iezzi, Carlos Murakami).
2. Videos: The "Matemática Rio com Prof. Rafael Procopio" channel on YouTube has various detailed and highly explanatory videos on the topic of inequalities.
3. Web page: Khan Academy has an entire module dedicated to first-degree inequalities and how to solve them step-by-step. The link is: https://pt.khanacademy.org/math/algebra/one-variable-linear-inequalities.

Practical Activity

Activity Title: The Inequality Challenge in Real-World Applications

Project Goal

The goal of this project is to apply the theory of first-degree inequalities to solve real-world problems.

Detailed Project Description

Groups of students will be responsible for investigating a real-world situation where inequalities are used to solve a problem or optimize a process. As an example, you can consider economics (defining minimum and maximum prices for a product), nutrition (caloric intake of a menu) or physics (resistance limits of a material under certain conditions).

After selecting the situation and obtaining the approval of the teacher, each group must develop a mathematical problem that must be solved through the use of first-degree inequalities. The students must identify the variables of the problem, formulate the inequality and only then solve it mathematically.

Required Materials

1. Books and/or internet resources for research and study of the topic.
2. Notebook or sheets of paper for notes and mathematical calculations.
3. Pens and pencils.

Detailed Step-by-Step Instructions to Carry Out the Activity

1. Form groups of 3 to 5 students.
2. Choose a real-world scenario where a first-degree inequality can be applied to solve a problem or optimize a process.
3. Write a detailed description of the chosen situation and identify the variables involved in the scenario.
4. Formulate a first-degree inequality that represents the problem or process.
5. Solve the inequality mathematically.
6. Write a final project report.

Project Deliverables and Written Document

The group must submit a written report in the following format:

1. Introduction: Describe the real-world situation you chose and the importance of inequalities for solving the problem or optimizing the process in the chosen scenario.
2. Development: Explain the theory of first-degree inequalities and how it applies in the chosen scenario. Describe the steps on how you formulated the inequality and how you solved it mathematically.
3. Conclusions: Recap the work done, indicate the knowledge acquired and the conclusions drawn about the project. What difficulties did you encounter? How did you overcome them? How did solving the inequality help you better understand the scenario?
4. Bibliography: Indicate the sources you consulted to work on the project, such as books, web pages, videos, etc.

The report should be submitted together with an oral presentation to the class, clearly and succinctly explaining the problem, the inequality, the solution and the conclusion.

This project should be completed within a one-month period, dedicating between five and ten hours of work per student.