Problem 5 - Fundamentals of Mathematics

Question

Write the appropriate symbol \((<,>,=)\) in place of the \(\square\) for each pair of numbers. $$ -1 \square-2 $$

Step-by-Step Solution

Answer

-1 > -2

Step 1: Understand the Concept of Inequalities

Inequality symbols are used to compare numbers. The symbol \(<\) means "less than," \(>\) means "greater than," and \(=\) means "equal to." When given two numbers, we need to determine which symbol correctly describes their relationship.

Step 2: Compare the Given Numbers

We have the numbers \(-1\) and \(-2\). In the number line, numbers to the right are greater than numbers to the left. Since \(-1\) is to the right of \(-2\) on the number line, \(-1\) is greater than \(-2\).

Step 3: Choose the Correct Symbol

Since \(-1\) is greater than \(-2\), the correct inequality sign to use is \(>\). Therefore, \(-1 > -2\).

Key Concepts

Understanding the Number Line

A number line is a visual tool that helps us understand the relative values of numbers. Imagine a horizontal line where every point corresponds to a number. These points extend infinitely in both directions. On this line, positive numbers are typically placed to the right of zero, while negative numbers are to the left.

When we place two numbers on a number line, their position tells us about their size compared to each other. For example, if you compare -1 and -2, you'll place these numbers on the left side of zero. -1 is positioned to the right of -2, indicating that it is larger or greater. Therefore, the number line not only shows the size of numbers but also helps clarify which number is bigger or smaller.

Using a number line can be advantageous when solving inequalities as it visually displays the difference between values, making it easier to compare numbers correctly.

Comparing Numbers

Comparing numbers involves determining if one number is larger, smaller, or equal to another. This is a crucial skill in understanding and solving problems involving inequalities.

Here's how you can compare numbers effectively:

• **Identify the Numbers:** Write down the numbers you wish to compare. For instance, say you have -1 and -2.
• **Use a Number Line:** As described earlier, plot each number on a number line. -1 will be to the right of -2, indicating that -1 is greater.
• **Analyze Positioning:** Look at their relative positions. The number to the right is always greater, while the one to the left is smaller.

Once familiar with comparing numbers on a number line, you can easily solve inequalities by assessing the positioning of numbers against one another.

Inequality Symbols

Inequality symbols are an essential part of mathematics that help us communicate the relationship between two numbers. The most common inequality symbols include:

• Less Than ( < ): This symbol is used when the first number is smaller than the second. For example, -3 < -1.
• Greater Than ( > ): This symbol appears when the first number is larger than the second. For instance, -1 > -2.
• Equal To ( = ): Used when two numbers or expressions have the same value. For example, 2 = 2.

In the original exercise, when we compare -1 and -2, we found that -1 is greater than -2 by using a number line. Therefore, we use the inequality symbol '>' to express this relationship. Understanding these symbols and their meanings is key to working with inequalities effectively.