Inequalities: Introduction

Relevance of the Topic

Inequalities, a natural extension of equations, are key to understanding how mathematical relationships can be unequal. They bridge the gap to future concepts such as linear functions and systems of inequalities, and have practical applications in many disciplines, including physics, economics, and engineering. Learning to solve them allows you to effectively understand and represent a wider variety of mathematical situations.

Contextualization

If we imagine mathematics as a large building, inequalities are one of the fundamental foundations. This is the first step towards understanding how numbers can be related not only by equality but also by inequalities.

In the 7th-grade Mathematics curriculum, after the introduction to equations, students begin to explore inequalities, a concept that extends and deepens their previous understanding of real numbers. This topic serves as a starting point for learning more complex and challenging content in subsequent years.

By understanding inequalities, you develop not only your mathematical ability but also logical and analytical reasoning skills, as solving an inequality requires the ability to identify and correctly interpret numerical relationships.

Theoretical Development

Components

• Inequality: An inequality is a mathematical expression that contains inequality signs (>, <, ≥, ≤) and represents an inequality relationship between two expressions.

  • For example, in the inequality "3x + 2 < 7", we see that "3x + 2" and "7" are related by an inequality, in this case, the "less than" sign (<).

• Variable: The variable, usually represented by "x", is an unknown element whose value can vary. In inequalities, we solve for the variable, determining which values of it satisfy the inequality.

  • In the above inequality, "x" is the variable, and we are looking for the values of "x" that make the expression "3x + 2" less than "7".

• Constants: Constants are known and fixed values. In inequalities, constants are present both in the expressions being compared and in the limits of the inequality.

  • In the given example, "3", "2", and "7" are constants.

Key Terms

• Literal Term: A literal term contains variables. In the example '3x+2<7', "3x" is a literal term.
• Isolated Term: An isolated term does not contain variables. In the example, "2" is an isolated term.
• Inequality Sign: It is a mathematical symbol that represents an inequality relationship. The main inequality signs are: greater than ( > ), less than ( < ), greater than or equal to ( ≥ ), less than or equal to ( ≤ ).

Examples and Cases

• Case 1: Consider the inequality '2x > 10'. To solve it, we start by isolating the variable "x". Dividing both sides of the inequality by 2, we get x > 5, indicating that all numbers greater than 5 are solutions to the inequality.

• Case 2: In the inequality '3x - 6 ≤ 9', we begin solving by adding 6 to both sides of the inequality, which gives us '3x ≤ 15'. Then, we divide both sides by 3, resulting in 'x ≤ 5'. This informs us that all numbers less than or equal to 5, including 5, are solutions to the inequality.

Both examples illustrate the application of the equality principle in inequalities: any operation we perform on an inequality must be done on both sides, keeping the inequality true.

Detailed Summary

Relevant Points:

• Inequalities are mathematical expressions that present inequality relationships between their parts. These relationships are expressed by mathematical signs, such as greater than (>) and less than (<).

• Inequalities involve not only the manipulation of numbers but also of variables. The solution to inequalities is the set of values that the variable can have so that the inequality is true.

• The equality principle also applies to inequalities. Any operation performed on an inequality must be done on both sides of the expression, keeping the inequality true.

• It is important to distinguish between literal terms (which contain variables) and isolated terms (which do not contain variables) in an inequality.

Conclusions:

• Solving inequalities is an essential skill in the study of mathematics, as it allows for the understanding of numerical relationships that are not only equal but also greater or lesser.

• The solution to an inequality is a set of numbers, not just a single value. This set can contain an infinity of values, or even none, depending on the inequality.

• The manipulation of terms and the correct application of mathematical principles are fundamental to solving inequalities properly.

Exercises:

1. Given the inequality '2x + 3 > 7', determine what values of 'x' satisfy the inequality. (Answer: x > 2)

2. Solve the inequality '5 - 3x ≥ 7'. (Answer: x ≤ -1)

3. Find the solution to the following inequality: '4x - 6 > 10 + 2x'. (Answer: x > 8)