Problem 4 - Precalculus Mathematics for Calculus

Question

The symbol \(|x|\) stands for the _____ of the number \(x\). If \(x\) is not \(0,\) then the sign of \(|x|\) is always _____.

Step-by-Step Solution

Answer

The symbol \(|x|\) stands for the absolute value of the number \(x\). If \(x\) is not 0, the sign of \(|x|\) is always positive.

Step 1: Understanding absolute value

The symbol \(|x|\) represents the absolute value of the number \(x\). Absolute value refers to the distance a number is from zero on the number line, without considering direction. This means \(|x|\) is always positive or zero.

Step 2: Determining the sign of the absolute value

Since the absolute value is the distance from zero, it is never negative. If \(x\) is not zero, then \(|x|\) is always positive.

Key Concepts

Understanding the Number Line

A number line is a simple visual tool that helps us understand the order and relative size of numbers. Imagine a straight line where each point corresponds to a number. At the center of this line is zero, often marked prominently. Positive numbers are placed to the right of zero, while negative numbers are to the left.

Number lines make it easy to visualize concepts like addition and subtraction, as well as the absolute value. When dealing with absolute values, we focus on the distance a number is from zero, without regard to direction. Because of this, even if a number is negative, its distance from zero — or its absolute value — is expressed as a positive number.

Exploring Positive Numbers

Positive numbers are numbers greater than zero. They are placed on the right side of the number line and have no negative sign. These numbers are straightforward because their absolute value is the number itself.

• Examples of positive numbers include 1, 2, 100, and so on.
• The absolute value of a positive number is the same as the number itself, making interpretation easy.

Positive numbers are intuitive to work with because they naturally express the concept of 'magnitude' or 'distance' when thinking of them as steps from zero on the number line.

The Role of Distance from Zero

The concept of distance from zero is crucial to understanding absolute value. This distance is always non-negative, meaning it can either be zero or a positive number. When a value is further from zero, its absolute value reflects this distance regardless of whether the original number is positive or negative.

• For a positive number, this distance is the number itself.
• For a negative number, this distance is its positive counterpart.
• For zero, the distance from zero is simply zero.

Absolute value is handy in real-world contexts where only the magnitude matters, not the direction — for example, when measuring the difference in temperature or the gap in height between objects.