Problem 39 - Prealgebra

Question

Find each of the following absolute values. $$|-8|$$

Step-by-Step Solution

Answer

The absolute value of \(-8\) is \(8\).

Step 1: Understanding Absolute Value

The absolute value of a number is the distance of the number from zero on the number line, regardless of direction. Absolute value is always non-negative. The absolute value of a number \(x\) is denoted by \(|x|\).

Step 2: Apply the Absolute Value Definition

For the number \(-8\), the absolute value is its distance from 0 on the number line. Since distance is always a positive quantity, \(|-8|\) is simply \(8\).

Step 3: Calculate the Absolute Value

By applying the understanding from the previous step, we find \(|-8| = 8\). This is because the distance from \(-8\) to \(0\) is \(8\) units.

Key Concepts

Understanding the Number Line

The number line is a powerful visual representation that helps us understand numbers better. Imagine a horizontal line with zero at its center. This is the number line. Numbers to the right of zero are positive, while those to the left are negative. Think of each mark on this line as a step or unit.

On the number line, the absolute position of any number shows how far it is from zero. This distance is recorded as an absolute value, which is always a non-negative number. For instance, the number \(8\) is 8 steps to the right of zero.

• Negative numbers lie to the left of zero.
• Positive numbers lie to the right of zero.
• The number zero itself has no direction but represents the center.

Understanding this simple layout of the number line can make problems related to absolute values and distances much easier to solve.

Defining Distance on the Number Line

Distance is a concept that often comes up when discussing the number line and absolute values. In real life, distance refers to how far apart two points are. On the number line, it's much the same. However, what’s interesting is that distance, like absolute value, is always a positive measurement.

To find the distance between two numbers on the number line, we observe how many steps are needed to move from one to the other, ignoring the direction. This means whether you're moving from a negative to a positive number or vice versa, the distance remains the same.

• Distance is always positive.
• Think of distance as the number of steps between two points.
• Direction doesn't affect the distance.

This is why when we find the absolute value of \(-8\), we calculate that it's 8 units away from zero, because distance is always non-negative.

Understanding Non-Negative Numbers

Non-negative numbers include all positive numbers and zero. The term "non-negative" means that the values are zero or greater, excluding any negative numbers. This applies directly to the concept of absolute value because absolute values are defined to be non-negative.

The idea that absolute value is non-negative stems from its definition, which is the distance from zero on the number line. Since distance can't be negative, neither can absolute values. Thus, \(|x|\) always results in a non-negative number, whether \(x\) itself is positive, negative, or zero.

• Non-negative numbers: positive numbers and zero.
• Absolute value is a perfect example of using non-negative numbers.
• The absolute value of any number is at least zero, and up to any positive value.

By focusing on non-negative numbers, we simplify calculations and understanding of concepts in mathematical exercises dealing with absolute value.