Problem 21 - Fundamentals of Mathematics

Question

For the pairs of real numbers in the following 5 problems, write the appropriate symbol \((<,>,=)\) in place of the \(\square\) $$ \begin{array}{lll} -7 & \square & -2 \end{array} $$

Step-by-Step Solution

Answer

-7 < -2

Step 1: Understand Number Line Position

The number line helps us visualize the order of real numbers. Numbers increase as you move from left to right. Thus, a smaller number will appear to the left of a larger number.

Step 2: Compare the Numbers

Let's compare the given numbers: - First number: -7 - Second number: -2 Because -7 is to the left of -2 on the number line, -7 is less than -2.

Step 3: Select the Appropriate Symbol

In mathematical notation, "less than" is represented by the symbol \(<\). Since -7 is less than -2, we replace the square with \(<\).

Key Concepts

Number Line

The number line is a powerful tool for understanding how numbers relate to each other. Imagine a straight line where every point represents a real number. This line stretches infinitely in both directions, making it easy to see where numbers are in relation to each other.

A few key points to remember about the number line:

• Numbers increase as you move from left to right.
• Zero is the central point, with negative numbers to the left and positive numbers to the right.
• The further left you go, the smaller the numbers become; the further right, the larger.

Using this visual tool, comparing numbers becomes straightforward. For example, if you were placing -7 and -2 on a number line, -7 would be further to the left than -2, indicating that -7 is smaller.

Ordering Numbers

Ordering numbers refers to arranging them from smallest to largest or vice versa. This process often involves comparing pairs of numbers to decide which comes first in the order.

Here’s a simple method to order numbers effectively:

• Place the numbers on a number line.
• Identify which numbers appear to the left and which to the right.
• Arrange them accordingly, from left (smallest) to right (largest).

Keeping this approach in mind helps in solving problems like the one in the exercise where we determined that -7 is less than -2 because it appears to the left on the number line. Thus, -7 would come before -2 in an ordered list from smallest to largest.

Inequalities

Inequalities describe the relative size or order of two numbers without giving their exact values. They use symbols like \(<\), \(>\), and \(=\) to show these relationships. In the exercise, we replaced a square with the appropriate inequality symbol to express that one number is less than, greater than, or equal to another.

Some key symbols and their meanings are:

• \(<\): Indicates that the number on the left is smaller than the number on the right.
• \(>\): Indicates that the number on the left is larger than the number on the right.
• \(=\): Indicates that both numbers are equal.

In our example, we found that -7 is less than -2, so we used the \(<\) symbol. Understanding and using inequalities is essential for solving mathematical problems that involve comparing numbers.