Contextualization

Introduction

Quadratic inequalities are a fundamental topic in the study of mathematics, more specifically in algebra. Similar to a quadratic equation, a quadratic inequality is a mathematical expression that involves a squared variable. The key difference is that instead of being equal to something (equation), in an inequality the expression is "greater than", "less than", "greater than or equal to", or "less than or equal to" something.

The structure of a quadratic inequality is ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0, where a, b, c are real numbers and a ≠ 0. Learning to solve this type of problems is essential, as many real-world situations can be modeled with the help of second-degree inequalities.

To solve quadratic inequalities, it is necessary to have a good understanding of quadratic equations, as solving the inequalities involves solving equivalent equations. The study also involves a graphical analysis, as it is common to visualize the solution through parabolic graphs.

Importance of Quadratic Inequalities

In the real world, mathematicians and scientists use inequalities in the development of mathematical models to study and solve practical problems. For example, in the field of economics, inequalities are often used in linear programming to optimize operations and increase efficiency. In physics or engineering, quadratic inequalities can be used to model the motion of an object in free fall and determine the maximum height it will reach or the time needed to reach the ground.

Furthermore, knowledge of quadratic inequalities is essential to advance in more complex mathematical studies, such as calculus. Therefore, mastering this topic is essential for anyone looking to pursue a career in science, technology, engineering, or mathematics.

Practical Activity

Activity Title: "Mastering Quadratic Inequality through Mathematical Modeling"

Project Objective

The purpose of this project is, through teamwork, to apply the concepts and techniques of quadratic inequalities in real-world situations, by building a mathematical model that will solve a proposed problem. During the project, it is expected that students will improve their problem-solving skills, teamwork, and time management.

Detailed Project Description

Groups of 3 to 5 students must choose a real-world situation that can be represented by a quadratic inequality. The goal is to model the situation using a quadratic inequality, solve the inequality, and interpret the result in the original context.

Examples of situations: choosing cell phone or broadband plans (based on minutes or data used), optimizing a production process, production and sales planning (based on fixed and variable costs), determining the trajectory of an object launched under certain conditions, among others.

The project should be completed in one week, and it is estimated that each group member will spend 2 to 4 hours working on the project.

Required Materials

• Paper, pencil, or pen to sketch and solve the quadratic inequality.
• Calculator, for calculations.
• Computer with internet access, for research and writing the final report.

Detailed Step-by-Step Guide for the Activity

1. Form groups of 3 to 5 students and choose a real-world situation that can be represented by a quadratic inequality.
2. Write a detailed description of the selected situation, explaining why it can be modeled by a quadratic inequality.
3. Modeling: Build the quadratic inequality that represents the chosen situation.
4. Solve the quadratic inequality, showing all the steps used in the resolution.
5. Interpret the result of solving the inequality in the context of the original problem.
6. Develop a written report following the proposed structure: Introduction, Development, Conclusions, and Bibliography used, incorporating the steps developed above.

Project Deliverables

At the end of the project, each group must deliver:

• Detailed description and justification of the chosen situation.
• Model of the quadratic inequality describing the situation.
• Solution of the inequality with detailed steps.
• Interpretation of the solution of the inequality in the light of the real situation.
• Written project report, which should include: introduction (with contextualization of the theme and project objective), development (with explanation of the activity, methodology, and discussion of the results), conclusion (revisiting key points and explaining the learnings and conclusions about the project), and bibliography.

Students must ensure that the written report complements the work done during the project, providing a clear view of their understanding of the topic of quadratic inequalities and their ability to apply it in a real context.