Problem 26 - Prealgebra and Introductory Algebra

Question

Place the correct symbol, \(<\) or \(>,\) between the two numbers. $$-6 \quad-1$$

Step-by-Step Solution

Answer

-6 < -1

Step 1: Identify the numbers

The numbers given for comparison are -6 and -1.

Step 2: Visualize on a number line

If we visualize these numbers on a number line, -6 would be to the left of -1. The direction of the number line is such that as we move to the right, the numbers increase. So, numbers which come first (or to the left) are smaller.

Step 3: Insert correct symbol

As -6 is to the left of -1 on number line, -6 is considered to be smaller than -1. Therefore, the appropriate symbol to insert between them is \(<\), making the answer \(-6 <-1\).

Key Concepts

Understanding the Number Line

A number line is a straight, horizontal line that shows numbers in their natural order. It's a handy visual tool for comparing values and understanding their relationships.

On a number line:

• Numbers increase as you move to the right.
• Numbers decrease as you move to the left.

Visualizing numbers on a number line helps in comparing their sizes. For example, if you look at -6 and -1, you'll find -6 placed to the left of -1. This means -6 is smaller than -1. This visualization makes understanding comparisons easier.

Exploring Negative Numbers

Negative numbers are like the opposites of positive numbers. They are found on the left side of zero on the number line. Negative numbers show values less than zero and are often encountered in situations like temperatures below freezing or depths below sea level.

Key points about negative numbers:

• The further left a number is on the number line, the smaller it is.
• Negative numbers are always less than positive numbers.

So, if we're comparing -6 and -1, -6 is smaller because it's more to the left on the number line compared to -1.

Using Mathematical Symbols

Mathematical symbols like \(<\) and \(>\) help in showing the relationship between numbers. These symbols make it easy to communicate whether numbers are greater than or less than each other.

Here's how to use these symbols:

• \(<\) means "is less than".
• \(>\) means "is greater than".

In the comparison of -6 and -1, the correct symbol is \(<\) because -6 is less than -1. This is written as \(-6 < -1\), clearly showing the relationship. This method is a clear and efficient way to represent numerical comparisons without the need for lengthy explanations.