Problem 52 - Introductory and Intermediate Algebra for College Students 4th

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$-1.25\quad\square-0.5$$

Step-by-Step Solution

Answer

-1.25 < -0.5

Step 1: Understanding Negative Numbers

Negative numbers are opposite in nature to positive numbers. On a number line, negative numbers are located to the left of zero and the farther left a number is, the smaller it is. Therefore, in the context of negative numbers, a number will be 'larger' if it is closer to zero. Hence, among negative numbers, -3 is smaller than -2, because -3 is farther from zero than -2.

Step 2: Comparing -1.25 and -0.5

-1.25 and -0.5 are both negative numbers. -0.5 is closer to zero than -1.25. Based on our understanding of negative numbers, -0.5 is 'larger' than -1.25 because it is closer to zero.

Step 3: Inserting the correct symbol

When comparing two numbers, the symbol '\(>\)' is used if the number on the left is greater than the number on the right, while the symbol '\(<\)' is used if the number on the left is smaller than the number on the right. Since -0.5 is greater than -1.25, the correct symbol to insert would be '\(<\)'.

Key Concepts

number line

The number line is a simple yet powerful tool to help us understand numbers and their relationships in a visual way. Imagine a straight line where each point corresponds to a number. The center of this line is zero, and all numbers to the right are positive, while those to the left are negative.

This line allows us to see the order of numbers easily. As you move to the right on the number line, the numbers increase in value. Conversely, as you move to the left, the numbers decrease. This becomes especially helpful when dealing with negative numbers because it shows how they relate to zero and to each other.

For example, on the number line, you will find -1.25 further left than -0.5, indicating that -0.5 is closer to zero. This provides a clear visual representation that helps us compare negative numbers efficiently.

inequality symbols

Inequality symbols such as ' < ' and ' > ' are critical in math to compare numbers and express their relationships. These symbols tell us which number is larger or smaller.

The ' < ' symbol means 'less than', while the ' > ' symbol means 'greater than'. For example, when you see -1.25 < -0.5, it tells you that -1.25 is less than -0.5. Conversely, -0.5 > -1.25 would indicate that -0.5 is greater than -1.25.

These symbols are not just useful for comparing negative numbers—they are used across mathematics wherever comparisons are necessary. Understanding and using inequality symbols correctly enables us to convey mathematical relationships clearly and effectively.

understanding negative numbers

Negative numbers might seem tricky at first because they behave differently than positive numbers. When you think about negative numbers, imagine them as being less than zero. Instead of adding value, they represent a deficit or a value that is less.

An important concept to grasp is that, with negative numbers, the number closer to zero is actually considered greater. For instance, while -1.25 may feel larger numerically, it is smaller than -0.5 because it is further from zero. This sometimes confuses students, but a strong mental picture of a number line can help.

• Negative numbers increase as they move towards zero on the number line.
• The closer a negative number is to zero, the 'larger' it is.

Remembering these points will help you confidently compare negative numbers in any mathematical context.