Problem 72 - Introductory Algebra for College Students

Question

Find each absolute value. $$|3|$$

Step-by-Step Solution

Answer

The absolute value of 3 is 3.

Step 1: Understand Absolute Value

The absolute value of a number is its distance from 0 on the number line, regardless of direction. It is always non-negative: \(|a| \geq 0\).

Step 2: Apply the Definition

We evaluate the absolute value of the given expression. Since the absolute value removes any negative sign, we take the non-negative value.

Step 3: State the Result

The absolute value of 3 is 3.

Key Concepts

Understanding the Number Line

The number line is a visual tool that helps us understand numbers in a straight line.
It includes all numbers, both positive and negative, arranged in increasing order from left to right.
The center of this line is zero.

• Numbers to the right of zero are positive.
• Numbers to the left of zero are negative.
• Each point on the line represents a unique number.

Understanding how numbers are placed on a number line is crucial, especially when dealing with concepts like absolute value.
It visually shows the distance a number is from zero, which is the core idea behind absolute value.

The Role of Positive Numbers

Positive numbers are essential for everyday calculations and understanding absolute value.
These are numbers greater than zero and are located to the right of zero on the number line.

• Examples include 1, 2, 3, and so on.
• They are used to represent quantities, distances, and magnitudes in real life.
• In calculations, they show gain or increase.

When we look at an absolute value, if the number is positive, it stays the same.
Understanding positive numbers helps clarify why \(|3| = 3\).
The distance is already positive and remains unaffected.

Understanding Distance from Zero

Distance from zero is a measure of how far a number is from the center of the number line.
Absolute value equations like \(|3|\) help capture this concept.

• The concept applies to both negative and positive numbers.
• For \(|3|\), the distance is directly the positive value itself, 3.
• Even if a number is negative, the absolute value gives its distance as a positive value.

Understanding distance from zero helps in solving real-world problems.
It may relate to anything needing a non-directional measure—like real-world distances or differences in quantities.
Knowing this concept aids in grasping that absolute values are non-negative, reflecting distances on a number line.