Problem 69 - Elementary and Intermediate Algebra

Question

Insert one of the symbols \(>,<,\) or \(=\) in the blank to make each statement true. $$ |3.4| \quad \text{_____}\quad-3 $$

Step-by-Step Solution

Answer

Insert \(>\) to make the statement true: \(|3.4| > -3\).

Step 1: Find the Absolute Value

The problem asks us to compare the absolute value of 3.4 and -3. First, calculate the absolute value of 3.4, which is the distance from 0 on the number line and is always non-negative. Thus, \(|3.4| = 3.4\).

Step 2: Evaluate the Negative Number

Recognize that -3 is already a number, and its position on the number line is left of 0. The value of -3 is -3.

Step 3: Compare the Numbers

Now, compare the two numbers: 3.4 (from the absolute value) and -3. Since 3.4 is to the right of -3 on the number line, it is greater than -3. Therefore, the correct symbol to insert to make the statement true is \(>\).

Key Concepts

Understanding the Number Line

A number line is a visual tool that allows us to see numbers in a linear sequence, where each point on the line corresponds to a number. It helps us understand concepts like absolute values, negative numbers, and comparisons between numbers.

Generally, a number line includes positive numbers to the right of zero and negative numbers to the left.

• The further right a number is, the larger it is.
• The further left you move, the smaller the number becomes.

Using a number line, you can easily identify which of two numbers is greater by looking at their positions. Numbers closer to the right are always larger. This concept is particularly useful when dealing with absolute values, as they represent distances from zero.

Comparing Numbers Using the Number Line

When comparing numbers, like in our exercise, we often use a number line to illustrate the comparison visually.

There are certain steps you can follow:

• Identify each number's position on the line.
• Recognize that a number on the right is greater than one on the left.

In the given problem, comparing \(|3.4|\) and \(-3\), we first locate these points on the number line. The absolute value \(3.4\) lies to the right of \(-3\).

This indicates that \(3.4\) is greater than \(-3\), leading us to use the symbol \(>\) for correct comparison. This method is straightforward once you familiarize yourself with reading and interpreting number lines.

Introduction to Negative Numbers

Negative numbers might initially seem confusing, particularly if you are more familiar with only positive numbers. However, they play an essential role in mathematics.

Negative numbers are found left of zero on the number line and represent values less than zero. Whenever a negative number like \(-3\) is placed on a number line, it is always to the left of any positive number and zero. Some key aspects of negative numbers include:

• They decrease as they move further left from zero.
• They are always less than their positive counterparts and zero.

Understanding where negative numbers fit on the number line will help you correctly compare them with other numbers, both positive and absolute values.