Problem 71 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The absolute value of 6 is 6.

Step 1: Understand the context of absolute value

An absolute value of a number is a nonnegative value that represents its distance from zero on a number line. It is denoted with the symbol '|' on either side of the number. Therefore, \( |6| \) represents the absolute value of 6.

Step 2: Determine the absolute value

Since 6 is a positive number and is 6 units away from 0 on the number line, its absolute value is 6 itself. Therefore, \( |6| = 6 \).

Key Concepts

Absolute Value Definition

When it comes to understanding numbers, the concept of absolute value is fundamental in algebra. The absolute value of a number is essentially its distance from zero on the number line, regardless of direction. In other words, it's a measure of how far a number is from zero, without considering whether it is to the left (negative) or to the right (positive).

Denoted by vertical bars on either side of the number—like this, \(|6|\)—the absolute value is always a nonnegative number. For instance, while -6 is six units to the left of zero, its absolute value, \(|-6|\), is 6, because distance cannot be negative. Similarly, the absolute value of 6 is 6, because it's also six units from zero but on the right side. Recognizing that absolute value is a type of distance can help one visualize and better understand many equations and inequalities in algebra.

Number Line

The number line is a visual representation of numbers laid out in a straight line, and it's a crucial tool in understanding mathematical concepts, including absolute value. On this line, zero is typically found at the center, with positive numbers extending indefinitely to the right, and negative numbers to the left. It's very much like a ruler measuring both sides from a central point.

When dealing with absolute values, the number line comes in handy to visually see the distance of a number from zero. To determine the absolute value, you simply count the number of units between the number and zero. For positive numbers like 6, it's straightforward because they already lie to the right of zero, hence \(|6|=6\). Seeing this visually can solidify the concept and its applications in arithmetic operations, graphing, and solving equations.

Positive Numbers

In the realm of mathematics, positive numbers are those that are greater than zero, and they fall to the right of zero on the number line. These numbers represent quantities that you can quantify or count, like the number of apples you might buy or the number of pages in a book.

A key property of positive numbers is that they remain the same when you take their absolute value. By their very nature, they are already at a 'positive distance' from zero. In our exercise, the number 6 is a positive number. Hence, when we seek the absolute value, we find that \(|6|=6\), as there is no 'negative' distance from zero in this case. Understanding positive numbers helps to understand the concept of absolute values better, especially when you need to compare and order different numbers.