Problem 71 - Introductory Algebra for College Students

Question

Find each absolute value. $$|6|$$

Step-by-Step Solution

Answer

The absolute value of 6 is 6.

Step 1: Understanding absolute value

An absolute value of a number can be thought of as its distance from zero on the number line. It is represented by the symbols | |. Absolute values are always non-negative. So, the absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart.

Step 2: Apply absolute value to the given number

Given number is 6. Since 6 is already a positive number, the absolute value is the number itself.

Key Concepts

Number Line

The number line is a visual representation used to compare and understand numbers. Imagine a straight line that extends infinitely in both directions. On this line, numbers are positioned in a sequential order.

• Zero is typically placed at the center.
• Positive numbers extend to the right of zero.
• Negative numbers extend to the left of zero.

This line helps illustrate the concept of absolute value. When we talk about absolute value, we are referring to the distance a number has from zero on this line. For example, on the number line, the number 6 is simply 6 units away from zero. The use of the number line makes it easier to see that distance indicators, such as absolute values, are non-negative.

Positive Numbers

Positive numbers are all the numbers greater than zero. They are situated to the right of zero on the number line. Some common characteristics include:

• They do not have any negative sign in front of them.
• Examples include 1, 2, 3, and so forth.
• They increase in value as they move further right on the number line.

When it comes to absolute value, a crucial point to note is that the absolute value of any positive number is the number itself. So, if we take the number 6, as in our example, its absolute value is simply 6. This is because it is already on the positive side with no distance added or subtracted from zero.

Mathematical Operations

Mathematical operations are fundamental actions taken on numbers. These include addition, subtraction, multiplication, and division. In the context of absolute value:

• Regard addition and subtraction as between distances on the number line.
• Multiplication and division, in contrast, change the magnitude of a number's distance.

Understanding these operations helps in problem-solving across various mathematical contexts. Absolute value is often integrated into these operations. For any absolute value task, especially with positive numbers, the goal is simplification.Consider the absolute value of 6, denoted as \(|6|\). It remains unaffected because the operation—applying absolute value—just confirms its innate distance from zero. This simplification demonstrates how these operations interlace with our number properties, helping solve more complex problems efficiently.