Problem 72 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The absolute value of 3 is 3.

Step 1: Interpret the Absolute Value

The absolute value of a number is that number's distance from zero on the number line. The absolute value of a number is always positive, because distances can't be negative.

Step 2: Calculate the Absolute Value

We want to find the absolute value of 3, denoted as \(|3|\). 3 is a positive number, so the absolute value of 3 is 3 itself.

Key Concepts

Understanding the Number Line

Imagine a straight line that extends infinitely in both directions. This is your number line. It is a helpful tool in mathematics for visualizing numbers and their positions relative to each other. The number line represents each number as a point along the line, where positive numbers are placed to the right of zero and negative numbers to the left.
On this line:

• Zero is the central point, acting as a reference.
• Positive numbers move to the right.
• Negative numbers move to the left.

This setup makes it easier to understand absolute value, as we can clearly see and measure the distance any number is from zero. For example, to find where the number 3 lies, you would simply move three steps to the right of zero.

The Role of Positive Numbers

Positive numbers are all the numbers greater than zero but not including zero itself. They are found on the right side of the number line. In real life, we encounter positive numbers in a variety of ways:

• Temperature above freezing
• Heights above sea level
• Money that you own

When dealing with absolute value, if you start with a positive number, the result remains the same. This is because absolute value measures how far a number is from zero, regardless of its direction. So if you find the absolute value of a positive number, like 3, you're simply stating that its distance from zero is 3.

Calculating Distance from Zero

Distance from zero is the core concept of absolute value. Regardless of a number's position on the number line, calculating its absolute value means determining how many steps it takes to reach zero.
Key points about distance from zero:

• Always a non-negative number.
• Absolute value of any positive number is the number itself.
• Absolute value of a negative number is its positive counterpart.

For the number 3, the distance from zero is simply 3 steps. This is why the absolute value of 3 is written as \(|3| = 3\). No matter if you deal with positive or negative numbers, absolute value helps simplify their distance from zero into a straightforward, positive measurement.