Problem 109 - Introductory Algebra for College Students

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. \(-24 \text { _____ }-20 \text { (Section } 1.3, \text { Example } 6)\)

Step-by-Step Solution

Answer

-24 < -20

Step 1 Understanding the Problem

Here, there are two numbers given, -24 and -20. The task is to compare these numbers. When we talk about negative numbers, the one that is farthest from zero on the negative side of number line is the smallest. So, compare these two numbers with respect to zero.

Step 2 Compare -24 and -20 to zero

-24 is farther from zero compared to -20 on the number line. Hence, -24 is less than -20.

Step 3 Insert the Correct symbol

Since -24 is less than -20, the correct symbol to insert is <.

Key Concepts

Negative Numbers

Negative numbers can sometimes be tricky because they behave differently than positive numbers. When we say a number is negative, that means it's less than zero. Imagine having a bank account; positive numbers mean money in the account, while negative numbers mean you owe money. The farther a negative number is from zero, the smaller its value.
For example, between -5 and -10:

• -10 is more negative, meaning smaller in value than -5.
• -5 is closer to zero and therefore greater than -10.

This concept can feel backward at first, but thinking about the number line helps make it clear.

Number Line

A number line is a great visual tool for understanding the size of numbers. It's a straight line where we plot numbers from left to right.

• Zero is at the center.
• Positive numbers go to the right.
• Negative numbers go to the left.

To find the value of a number, look at how far and in which direction it is from zero. The further left a number is, the smaller it is. For instance, on the number line:

• -20 is closer to zero than -24.
• Thus, -24 is smaller because it's placed further left.

Visualizing numbers on a number line makes it easier to see how numbers compare to each other.

Comparing Numbers

Comparing numbers involves determining which is bigger or smaller. With positive numbers, it's straightforward — larger means it’s bigger. But with negative numbers, you have to think in reverse.

• A larger negative number is actually smaller.
• To compare -24 and -20: think of which one is further from zero.

Since -24 is further from zero than -20, it is smaller. We express comparisons using inequality symbols such as "<" and ">", where less than is "<" and greater than is ">". Thus, -24 is less than -20, so we write:

• -24 < -20

By practicing with these comparisons, you'll become more comfortable with how negative numbers work.