Problem 51 - Introductory Algebra for College Students

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$-2.5 \square 1.5$$

Step-by-Step Solution

Answer

-2.5 < 1.5

Step 1: Understanding Negative Numbers

Firstly, it's important to understand the concept of negative numbers. On a number line, negative numbers are to the left of zero. This means that they are less than zero.

Step 2: Comparing Negative Number with Positive Number

-2.5 is a negative number and 1.5 is a positive number. We know from step 1 that negative numbers are less than positive numbers. Therefore, -2.5 is less than 1.5.

Step 3: Inserting the Correct Sign

Using the knowledge from step 2 that -2.5 is less than 1.5, we can insert the less than sign ( \(<\) ) in the shaded square to make a true statement meaning \(-2.5 < 1.5\).

Key Concepts

Negative Numbers

Negative numbers are interesting and important in mathematics as they expand our understanding of values below zero. When you work with negative numbers, think of them as the opposite of positive numbers. If you owe someone money, you might think of that as having a negative balance. Negative numbers appear to the left of zero on a number line, which is a visual way of representing these values.
On the number line, as you move further left, the value of the negative number becomes smaller. For instance,

• -5 is less than -3 because -5 is further to the left of -3 on the number line.
• Both -2.5 and -1.5 are negative, but -2.5 is further left than -1.5, making it smaller.

When comparing negative numbers to zero or positive numbers, it's crucial to remember that negative numbers are always less than zero or any positive number. This understanding is essential when solving inequalities involving negative numbers.

Number Line

The number line is a simple yet powerful tool for visualizing numbers, both positive and negative. Picture a horizontal line with arrows on both ends. It's endless, representing all the numbers in order. Zero is in the middle, acting as a separation point between negative numbers on the left and positive numbers on the right.
The number line allows us to see at a glance which numbers are larger or smaller. As you move from left to right on the number line:

• Numbers increase as you move towards the right, from negative, through zero, to positive numbers.
• Each step right represents an increase in value, while each step left represents a decrease.

This is particularly handy when comparing numbers. For example, if you are comparing -2.5 and 1.5 on the number line, -2.5 is on the left, while 1.5 is on the right, indicating -2.5 is less than 1.5. The number line helps to visually demonstrate why this inequality holds true.

Positive Numbers

Positive numbers represent quantities greater than zero and bring to mind things we often encounter like money in a piggy bank or scores in games. On the number line, positive numbers reside to the right of zero. This positioning helps us quickly understand that any positive number is larger than zero.
When we compare positive numbers:

• A higher number on the right is greater than one to its left. For example, 3 is greater than 2.
• If a positive number is compared with a negative number, the positive number is always greater.

For instance, 1.5, a positive number, is more than -2.5, a negative number. In this situation, checking on the number line immediately shows that positive numbers are farther right than negative numbers, reinforcing that positive numbers are always greater. Understanding positive numbers' place on the number line is crucial for making comparisons and solving inequalities effectively.