Problem 48 - 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. $$3\quad\square\quad\frac{3}{2}$$

Step-by-Step Solution

Answer

The correct symbol to place in the shaded area is \( > \). Thus, the true statement is: 3 \( > \) \( \frac{3}{2} \)

Step 1: Convert the integer into a fraction

The integer 3 can be converted into a fraction where the denominator is the same as the denominator of the other fraction. It becomes \( \frac{6}{2} \), which equals 3.

Step 2: Compare the fractions

A comparison between \( \frac{6}{2} \) and \( \frac{3}{2} \) shows that \( \frac{6}{2} \) is greater than \( \frac{3}{2} \).

Step 3: Choose the appropriate symbol

Since 3 (or \( \frac{6}{2} \)) is greater than \( \frac{3}{2} \), the appropriate symbol to insert would be \( > \).

Key Concepts

Fraction Conversion

Fractions represent parts of a whole, and they are essential components in mathematics, particularly when it comes to comparing numbers.

When we need to compare two numbers and one or both of them are fractions, it's often helpful to convert them to a common format. Converting integers to fractions, as seen in the example where we turned 3 into \( \frac{6}{2} \), can make it easier to compare them directly. This process involves choosing a common denominator, which is usually the denominator of the other fraction in the comparison.

Here's a simplified guide on how to convert an integer to a fraction that can be compared with another fraction:

• Identify the denominator of the fractional number you want to compare with.
• Use this denominator as the denominator for your new fraction.
• Multiply the integer by the denominator to find the new numerator.

Using this method ensures that we are comparing like with like, making it easier to see which number is larger or smaller.

Inequalities

Inequalities are mathematical expressions that show the relation of two values, indicating whether one is greater than, less than, or equal to the other. They are symbolized by signs such as greater than (>) or less than (<).

Understanding how to determine the correct inequality symbol is a foundational skill in math. To establish which symbol to use, we compare the values of the numbers or expressions on either side. In the example of comparing \( \frac{6}{2} \) and \( \frac{3}{2} \), we need to determine which fraction represents the larger quantity.

Here's how to evaluate inequalities involving fractions:

• Ensure both fractions have a common denominator.
• Compare the numerators, since the denominators are the same.
• The fraction with the larger numerator is the greater value.

Selecting the correct symbol will allow us to accurately convey the relationship between the two values.

Mathematical Symbols

Mathematical symbols are the shorthand of mathematics. They communicate complex ideas succinctly and precisely. The symbols for inequality, greater than (>) and less than (<), help us quickly understand the relationship between two values.

In the context of comparing fractions, knowledge of these symbols is critical. After analyzing the fractions and deciding which is larger or smaller, we use these symbols to record our conclusion. The choice between > and < hinges on whether the number on the left side of the symbol is larger or smaller than the number on the right.

Here's a tip to remember which symbol to use:

• The symbol > means greater than, and it opens towards the larger value.
• The symbol < means less than, and it points to the smaller value.

These symbols are fundamental to expressing mathematical relationships and their correct use is key to clear mathematical communication.