Problem 68 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$|-456|$$

Step-by-Step Solution

Answer

456

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, absolute value is always a positive number or zero.

Step 2: Apply the Concept

Given the number \(-456\), the absolute value is the positive form of the number since absolute value represents distance. So, \(|-456| = 456\).

Key Concepts

distance from zero

When we talk about the 'absolute value' of a number, we are essentially referring to its distance from zero on the number line. Distance, in this context, is always a positive measure. Imagine you are standing at zero on the number line: to calculate the absolute value of any number, you measure how far that number is from zero.
Examples:

For 5, the distance from zero is 5 units.

For -5, the distance from zero is also 5 units.

If you think about the concept of distance, it is never negative. You can't travel a negative distance, right? Therefore, the absolute value of any number is always non-negative.

positive number

A key feature of absolute value is that it always results in a positive number or zero. This is because we are measuring how far a number is from zero, and distance cannot be negative. For instance, whether you start from -456 or +456, the absolute value will treat them the same way, turning them into positive distances. The rule to remember is:

• If a number is negative, its absolute value is the positive counterpart. For example, \(-456\rightarrow 456\)
• If a number is already positive, its absolute value remains the same.
• The absolute value of zero is simply zero.

We translate any negative numbers to their positive equivalents when dealing with absolute values. This ensures that the results are always non-negative.

number line

Visualizing absolute values is easier when using a number line. A number line is a straight line with numbers placed at equal intervals along its length. It usually has zero in the middle, with positive numbers to the right and negative numbers to the left. When locating the absolute value of a number on the number line, you can see the 'distance from zero' directly.
Steps to Find Absolute Value on a Number Line:

Start at zero.

Move to the location of the number whose absolute value you want to find (e.g., -456).

Count the units back to zero.

The number you count represents the absolute value. For \(-456\), count 456 units back to zero, giving \(|-456| = 456\). The number line helps visualize why absolute values are positive.