Problem 26 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ |-4| $$

Step-by-Step Solution

Answer

The absolute value of \(-4\) is \(4\).

Step 1: Understand The Absolute Value

The absolute value of a number \(a\) is denoted as \(|a|\), can be thought as its distance from zero on the number line. This means it is always nonnegative.

Step 2: Apply the Concept to Our Problem

In this exercise for the absolute value of \(-4\), we have \(|-4|\). Considering the definition from step 1, the absolute value of a negative number makes it positive.

Step 3: Evaluate the Absolute Value

The absolute value of \(-4\) is \(4\), because \(4\) is \(4\) units away from zero on the number line which is the same distance as \(-4\), but in the positive direction.

Key Concepts

Understanding the Number Line

The number line is a visual representation of numbers in a straight line. Each point on the line corresponds to a number. The middle point, called zero, allows you to understand where positive and negative numbers are positioned.
On the number line, numbers to the right of zero are positive, and those to the left are negative. When we talk about the absolute value, it's helpful to see the number line as a tool that shows how far numbers are from zero, regardless of direction.

• Positive numbers move right from zero.
• Negative numbers move left from zero.
• Zero itself is the starting point for measuring distance on the line.

Using the number line, you can understand that both \(-4\) and \(4\) are equidistant from zero, reinforcing the concept of absolute value.

Evaluating the Expression

To evaluate an expression with absolute value, like \(|-4|\), we first need to grasp what the absolute value signifies. It asks for the distance a number is from zero, not taking into account its direction.
First, identify the number within the absolute value bars. Here, it's \(-4\).
Then, apply the absolute value rule:

• Ignore the negative sign.
• Recognize that the result is always nonnegative.

Therefore, when we evaluate \(|-4|\), we find it equals \(4\) because absolute value measures distance, which is always a positive quantity.

Calculating Distance from Zero

The concept "distance from zero" is crucial when dealing with absolute values. Distance, in this context, refers to how many units a number is from zero, without considering direction.
This means that whether you're \(-4\) units or \(4\) units from zero, the distance remains the same: it is \(4\) units.
Let's break this down:

• \(-4\), located to the left of zero, is four steps away from zero.
• \(4\), located to the right, is similarly four steps away.

Ultimately, the term "distance" helps us focus on how far a value is from zero, leading to a better understanding of absolute value as always positive and quantifiable.