Introduction

Relevance of the Topic

Modular Inequality is one of the fundamental components of mathematics. It allows solving problems that involve the distance between numbers in relation to a fixed point, called the modulus. This concept is vital for understanding later topics, such as exponential and logarithmic equations.

Contextualization

Within the vast field of mathematics, Modular Inequality is situated in the subsystem of equations and inequalities. It is a natural development from understanding linear equations and inequalities. Its introduction comes after mastering these basic concepts, marking the evolution of students' perception of the functioning of the number system. In the 1st year of high school, Modular Inequality is an important step in students' transition from the pure mathematics of elementary school to the advanced concepts to be addressed in the following years. Understanding Modular Inequality also opens doors to important concepts in other disciplines, such as Physics and Engineering.

Theoretical Development

Components

• Modulus (or Absolute Value): The modulus of a number, represented by |x|, is the same as the absolute value of x. This means that the modulus always returns a non-negative number. It indicates the distance that a number x is from zero on a number line.

• Modular Inequality: An inequality with modulus is one that involves the expression |f(x)| and a relationship of inequality, usually greater than (>) or less than (<). Solving this inequality involves finding the regions of the domain where the inequality is true.

• Solutions of Modular Inequality: The solution to a modulus inequality involves two cases: if the absolute value is positive (|f(x)| > a), the solution is the set of values of x that make f(x) greater than a or less than -a; if the absolute value is negative (|f(x)| < a), the solution is the interval in which the function f(x) is strictly between -a and a.

Key Terms

• Absolute Value: Represents the distance of a number from the origin on a number line. In modulus, it will always return a positive number or zero, never negative.

• Inequality: An inequality is a mathematical sentence that compares two expressions, generally using the symbols of equality (=), greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤).

• Variable: It is a symbol (usually a letter) that represents a fixed but unknown value. The value of the variable can change when the equation is solved.

Examples and Cases

• Case 1: Inequality with Isolated Modulus (|x| > 3): Solving this inequality, we have that x is greater than 3 or less than -3. This is because the modulus of any number is always positive (or zero). So, when the modulus is greater than 3, the number itself can be greater than 3 or less than -3.

• Case 2: Inequality with Function in Modulus (|2x-1| < 9): Solving this inequality, we have that the function 2x-1 is strictly between -9 and 9. Rearranging the inequality, we have -9 < 2x-1 < 9. Adding 1 to all parts of the inequality, we get -8 < 2x < 10. Dividing by 2, we find -4 < x < 5. So, the solution to the inequality is x belonging to the open interval (-4, 5).

These cases illustrate the diversity of Modular Inequalities and how they can be applied to different types of expressions. The ability to solve these inequalities is vital for understanding and applying mathematics in a broader context.

Detailed Summary

Key Points

• Meaning of Modulus: The absolute value (modulus) of a number represents its distance from the origin on a number line. It always results in a positive value or zero.

• Inequality vs Equation: An equation represents equality, while the inequality compares two expressions, indicating if they are equal or if one is greater/less than the other.

• Modulus in Inequality: The presence of the modulus in an inequality introduces a situation where the result can be positive or negative, depending on the value being compared.

• Operating with Modular Inequalities: It is essential to understand that the method for solving inequalities with modulus varies with the type of inequality (>, <, ≥, ≤) and the sign of the constant term, when present.

Conclusions

• Diversity of Solutions: Modular inequalities can have a variety of solutions, including sets of intervals, combinations of intervals, and empty sets.

• Practical Importance: The ability to solve inequalities with modulus is crucial for solving problems that involve distances, or differences, between measurements, numbers, or variables.

• Preparation for Future Topics: Mastering modular inequalities is an important step in preparing for more advanced concepts, such as exponential, logarithmic, and rational equations, which often include expressions with modulus.

Suggested Exercises

1. Solve the inequality |5-3x| > 10.

2. Determine all values of x that satisfy the inequality |2x-5| < 4.

3. Find the solution set of the inequality |4x-3| > -9.