Problem 17 - Calculus: One and Several Variables

Question

Evaluate. |6|.

Step-by-Step Solution

Answer

The evaluated absolute value of |6| is 6.

Step 1: Understand the absolute value

The symbol |.| represents absolute value. This means to get a positive number regardless of whether the entry is negative or positive. For instance, |6| is 6 and |-6| is also 6. In this case, the task is to evaluate |6|.

Step 2: Evaluate the absolute value of the number

According to the definition of absolute value, |6| would equate to 6 since 6 is already a positive number.

Key Concepts

Understanding Positive Numbers

Positive numbers are numbers greater than zero. They are the building blocks in learning about absolute values and many other mathematical concepts.
Positive numbers are typically considered "normal" numbers in everyday use. They indicate quantities or magnitudes. When we talk about positive numbers:

• They are to the right of zero on a number line.
• They include fractions, decimals, and whole numbers, as long as they are greater than zero.
• They often represent things that can be counted, like apples in a basket or money in a wallet.

Understanding positive numbers is crucial as they often serve as the direct "output" of evaluating absolute values.

Evaluating Expressions with Absolute Values

Evaluating expressions involving absolute values is a key skill. The absolute value is represented by vertical bars, for example, \(|x|\). It describes the distance of a number from zero on a number line.
This distance is always positive.
Here's how you evaluate absolute value expressions:

• If the number inside the absolute value bars is positive or zero, the result is the number itself. For example, \(|6| = 6\).
• If the number inside the absolute value bars is negative, remove the negative sign, making it positive. For example, \(|-3| = 3\).

The principle behind absolute values: no matter the original sign of the number, the result is always non-negative. Even when dealing with equations, the process of evaluating absolute values simplifies complex expressions by focusing on the magnitude rather than the direction of the number.

Essential Features of Mathematical Notation

Mathematical notation serves as the language of mathematics. It provides a concise and precise way to express mathematical ideas and relationships.
Understanding this notation is crucial for topics like absolute value and beyond.Key elements of mathematical notation include:

• Symbols: Mathematical operations are represented with symbols like +, -, *, and /, while special concepts use unique symbols, like the absolute value bars |.| for indicating absolute values.
• Equations: Equations use symbols and numbers to represent relationships, such as \(|x| = x\) for any non-negative x.
• Variables: Letters such as x or y often denote unknowns or variables, allowing for generalization of expressions and equations.

Equipping yourself with the ability to read and interpret these symbols is foundational for solving mathematical problems effectively. With practice, these symbols become tools that enhance understanding and problem-solving abilities.