91Ó°ÊÓ

Real numbers encompass a broad class of numbers that we encounter in everyday life. They include integers, fractions, and irrational numbers. Essentially, any number you can think of that exists on the number line is a real number. Examples of real numbers include:

• Whole numbers like 0, 1, 2, and so on.
• Fractions such as 1/2 or 3/4.
• Irrational numbers like \( \pi \) or the square root of 2.
• Negative numbers like -1, -2.5, etc.

Real numbers can be both positive and negative and can be expressed in decimal form as finite, repeating, or non-repeating sequences. Understanding real numbers is fundamental because they serve as the basis for more complex mathematical concepts, including inequalities.

When dealing with inequalities, we often encounter the concept of unions. The union of inequalities signifies combining two or more conditions, showing that a solution satisfies at least one of these conditions. In mathematical notation, the union is represented by the symbol \( \cup \).

For instance, if we have the condition \( x \leq 5 \) or \( x > 10 \), they form a union of inequalities written as \( x \leq 5 \cup x > 10 \). This essentially means that a real number \( x \) can either be:

• Less than or equal to 5, or
• Greater than 10

This combination allows us to represent ranges of values effectively and plays a crucial role in solving real-world problems that involve conditions not satisfied by a single inequality.

Inequality conditions represent the relationships between values, indicating how one value compares to another. These conditions are expressed using symbols such as:

• \( \leq \): Less than or equal to
• \( < \): Less than
• \( \geq \): Greater than or equal to
• \( > \): Greater than

In the context of our exercise, two inequality conditions are stated: "less than or equal to 5" and "greater than 10." Each condition describes a specific range of values in relation to a given number, aiding us in defining boundaries for solutions.

Understanding how to interpret and manipulate these inequality conditions is essential in various fields, such as research, engineering, and everyday decision-making. By mastering how to work with inequalities, we can define specific criteria that help characterize real-world scenarios effectively.