Problem 65 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$|5.6|$$

Step-by-Step Solution

Answer

The absolute value of 5.6 is 5.6.

Step 1: Understanding Absolute Value

Absolute value refers to the non-negative value of a number without regard to its sign. It is the distance of a number from zero on the number line.

Step 2: Identify the Number

Identify the number inside the absolute value symbols. Here, the number is 5.6.

Step 3: Apply Absolute Value Rule

Since 5.6 is already a positive number, its absolute value is 5.6.

Key Concepts

The Concept of Non-Negative Value

In mathematics, a non-negative value is a number that is greater than or equal to zero. These values are either positive numbers or zero itself.
Non-negative values include:

• All positive numbers, like 1, 2, 3, etc.
• Zero, which is neither positive nor negative
• Decimal numbers and fractions greater than zero, such as 0.5, 1.75, etc.

Understanding non-negative values is essential when working with absolute values, as the absolute value of any number is always non-negative. This means that the result of an absolute value function can never be negative since it represents a distance, which can't be negative.

Number Line Distance and Absolute Value

Absolute value can be thought of as the distance a number is from zero on a number line. Imagine walking from zero to another number.
For example:

• To find the absolute value of 5.6, consider its position on the number line. It’s 5.6 units away from zero.
• If the number was -5.6 instead, you would still walk 5.6 units to reach zero.

In both cases, regardless of direction, you're just measuring distance. Hence, absolute value strips away the sign of the number, focusing only on how far the number is from zero. This is why the absolute value of both 5.6 and -5.6 is simply 5.6.

Understanding Positive Numbers

Positive numbers are numbers greater than zero. They can be found to the right of zero on a number line:
This includes:

• Whole numbers like 1, 2, 3, etc.
• Decimal numbers like 0.1, 2.5, etc.

In the context of absolute value:

• If a number is already positive, its absolute value remains the same. So, \(|5.6| = 5.6\).
• Positive numbers represent quantities or measures above a baseline (zero).

This understanding makes it clear why the absolute value of any positive number is the number itself. Positive numbers naturally align with the concept of non-negative values and measuring distance, key aspects of absolute value.