Problem 48 - Introductory Algebra for College Students

Question

Insert either \(<\) or \(>\) in the shaded area between each pair of numbers to make a true statement. $$3 \square -\frac{3}{2}$$

Step-by-Step Solution

Answer

So, the correct symbol to insert in the shaded area is \(>\). The final statement becomes: \(3 > -\frac{3}{2}\).

Step 1: Placement on the Number Line

Place the two numbers on the number line. On a number line, the numbers to the right are greater than the numbers to the left. The number 3 falls to the right of \(-\frac{3}{2}\).

Step 2: Comparison

Since 3 is to the right of \(-\frac{3}{2}\) on the number line, it is greater than \(-\frac{3}{2}\).

Key Concepts

number line

Understanding the concept of a number line can be very helpful when comparing numbers. A number line is a straight line where numbers are placed according to their value. Each position on the line corresponds to a numeric value. Typically, numbers increase as you move from left to right.

Here's how you can visualize it:

• The center often marks zero (0).
• Positive numbers are placed to the right of zero.
• Negative numbers are placed to the left of zero.

By visualizing both numbers on a number line, it is much easier to see which number is bigger, especially when working with fractions or negative numbers. This visual aid helps us understand that any number located to the right of another number on this line is larger.

greater than

When comparing two numbers to determine which is larger, we use the term 'greater than.' The symbol for greater than is \(>\).

As a rule of thumb:

• If a number is to the right of another number on the number line, it is greater than that number.
• The number 3 is greater than \-\frac{3}{2}\ because it appears to the right on the number line.

Let's remember that this concept helps us compare both positive and negative numbers, whole numbers, and even fractions efficiently.

integer comparison

Integer comparison involves determining the order of whole numbers, both negative and positive. Integers are all the numbers on the number line without any decimal or fractional component.

To compare integers, you can use these steps:

• Plot both numbers on the number line.
• Observe which number appears further to the right or left.
• Decide the relationship—using less than (\(<\)) or greater than (\(>\)).

For instance, in comparing 3 and \-\frac{3}{2}\, we see that 3 is an integer situated rightward, indicating it is greater. This systematic approach ensures accuracy, especially when dealing with a mix of positive and negative integers.