Problem 65 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make a true statement. See Examples 6 through 8 . $$ \frac{18}{3} \quad \frac{24}{3} $$

Step-by-Step Solution

Answer

\(\frac{18}{3} < \frac{24}{3}\).

Step 1: Simplify the first fraction

Let's simplify \(\frac{18}{3}\). Divide 18 by 3: \[\frac{18}{3} = 6\]

Step 2: Simplify the second fraction

Now, simplify \(\frac{24}{3}\). Divide 24 by 3: \[\frac{24}{3} = 8\]

Step 3: Compare the simplified numbers

We have simplified the fractions to 6 and 8. Now compare 6 and 8: 6 < 8.

Key Concepts

Fraction Simplification: Breaking Down Fractions Easily

When dealing with fractions, a helpful first step is simplification. This process involves reducing a fraction to its simplest form by dividing the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD).
For example, in the fractions \(\frac{18}{3}\) and \(\frac{24}{3}\), both numerators (18 and 24) can be divided by the common denominator, which is 3.

• \(\frac{18}{3}\) can be simplified by dividing both 18 and 3 by 3, resulting in 6.
• Similarly, \(\frac{24}{3}\) simplifies to 8 when both 24 and 3 are divided by 3.

By simplifying, we convert both fractions into whole numbers, making them easier to compare directly.Understanding how to simplify fractions is crucial for accurate calculations in mathematics.

Inequality Symbols: Understanding Less Than, Greater Than, and Equal To

In mathematics, inequality symbols are used to compare the size or value of two numbers or expressions.
Here, the key symbols are:

• \(>\) means 'greater than'
• \(<\) means 'less than'
• \(=\) indicates 'equal to'

These symbols help us communicate which of two numbers is larger or smaller. For instance, comparing the simplified fractions from earlier, 6 and 8, requires using an inequality symbol. Since 6 is less than 8, we use \(<\).
This expression, 6 < 8, clearly tells us that 6 is smaller. Mastering these symbols is essential for solving many types of mathematical problems.

Mathematical Comparison: Making True Statements with Numbers

To make correct mathematical comparisons, it's crucial to first simplify any fractions involved, as done in our example. Once simplified, comparing numbers becomes straightforward. In our case, we have two numbers: 6 and 8.
Here's how to approach such comparison problems:

• Simplify both expressions, if necessary. This step clarifies what you are comparing.
• Identify the correct inequality symbol to express the relationship.
• Use the appropriate symbol to write the comparison. In our case, 6 < 8.

This method ensures that your solution is both accurate and clear, helping you confidently solve similar problems in the future. Understanding these steps makes tackling inequality problems much simpler.