Problem 59 - Beginning and Intermediate Algebra

Question

Find each absolute value. \(|6-3|\)

Step-by-Step Solution

Answer

\( |6 - 3| = 3 \)

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. The absolute value is always non-negative.

Step 2: Simplify Inside the Absolute Value

Simplify the expression inside the absolute value: \( |6 - 3| = |3| \)

Step 3: Evaluate the Absolute Value

Determine the absolute value of 3. Since 3 is 3 units away from zero, \( |3| = 3 \)

Key Concepts

Distance from Zero

Understanding the concept of distance from zero makes it easier to work with absolute values.
The absolute value of a number describes how far that number is from zero on the number line, regardless of direction. This means it doesn't matter if the number is negative or positive.
For example, the numbers 3 and -3 are both three units away from zero. Therefore, their absolute values are the same: \( |3| = 3 \) and \( |-3| = 3 \). The absolute value simply tells you the magnitude of the number without considering its sign.
As long as you remember that absolute value is all about distance from zero, you will find it easier to solve problems involving this concept.

Simplification

Simplifying the expression inside the absolute value is a crucial step in solving absolute value problems.
You need to perform any arithmetic operations contained within the absolute value symbols first.
Let's look at the example \( |6 - 3| \).

• First, solve the subtraction inside the absolute value: \( |6 - 3| = |3| \)
• Next, evaluate the absolute value of the resulting number, which is already simplified to \( |3| \).

Breaking down the problem into smaller steps makes it more manageable and prevents mistakes.
Always ensure the problem inside the absolute value is fully simplified before moving forward.

Non-Negative Number

Absolute values are always non-negative numbers.
This is because they represent distances, and distances cannot be negative.
For example, \( |3| = 3 \) and \( |-3| = 3 \). Both results are non-negative because we are dealing with distance from zero.
Keep in mind that even if the number inside the absolute value is negative, the result will still be positive or zero.
Absolute values transform any number into a non-negative value, providing a clear understanding of magnitude without the sign.