Problem 9 - Fundamentals of Mathematics

Question

Use the algebraic definition of absolute value to find the following values. $$ |-12| $$

Step-by-Step Solution

Answer

The value of \(|-12|\) is 12.

Step 1: Understand the Definition of Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. Thus, it is always non-negative.

Step 2: Apply the Absolute Value Rule

The algebraic definition of absolute value is as follows: - If the number is positive or zero, its absolute value is the number itself. - If the number is negative, its absolute value is the number multiplied by -1.

Step 3: Evaluate the Expression |-12|

Since -12 is a negative number, we use the second rule of absolute value. Multiply -12 by -1 to determine the absolute value. Thus, \(|-12| = -1 imes -12 = 12\).

Key Concepts

Number Line

The number line is a visual representation of numbers laid out in a straight line. It helps us understand relationships between numbers, especially useful when discussing the concept of absolute value.

On the number line, every point corresponds to a number. Numbers increase as you move to the right and decrease as you move to the left. Zero is positioned at the center, acting as a reference point.

The absolute value can be thought of as the distance a number is from zero on this line.

• No matter if a number is positive or negative, its absolute value is its distance from zero, never its direction.
• For example, the number -12 is twelve units away from zero. So, \(-12\) on the number line would mirror twelve places to the right of zero, just like positive 12.

Algebraic Definition

The algebraic definition of absolute value provides a precise method to determine the absolute value through basic rules.

This definition states that for any real number \(x\):

• If \(x\) is positive or zero, then \(|x| = x\).
• If \(x\) is negative, then \(|x| = -x\).

This rule simplifies finding absolute values and underscores the fact that absolute value is inherently non-negative. For example, the number -12 is negative, so its absolute value is computed by multiplying it by -1. Thus, \(|-12| = 12\). This clear-cut approach makes it easy to evaluate absolute values algebraically.

Non-Negative Numbers

Non-negative numbers are those that are zero or positive. They form an essential part of understanding absolute value.

Since the absolute value measures the distance on the number line without considering direction, absolute values are always non-negative.

Kindly note:

• Even if the original number is negative, like -12, its absolute value is a non-negative number, 12.
• Absolute values help in simplifying expressions and solving equations by providing consistency, as they ensure results are non-negative.

Understanding that absolute values yield non-negative results is crucial for several mathematical operations and real-world applications where only the magnitude matters, not the direction.