Problem 19 - Algebra A Combined Function

Question

Determine whether each statement is true or false. See Examples 1 through 6 and 10. $$ \frac{9}{10} \leq \frac{8}{9} $$

Step-by-Step Solution

Answer

The statement is false; \(\frac{9}{10} \leq \frac{8}{9}\) is incorrect.

Step 1: Understand the Problem

You need to determine if the fraction \( \frac{9}{10} \) is less than or equal to the fraction \( \frac{8}{9} \). This means checking whether \( \frac{9}{10} \leq \frac{8}{9} \) is a true or false statement.

Step 2: Compare Fractions

A common method to compare fractions is to cross-multiply to avoid dealing with decimals. That means comparing \(\frac{9}{10}\) and \(\frac{8}{9}\) by checking if \(9 \times 9 \leq 8 \times 10\).

Step 3: Perform the Cross-Multiplication and Compare

Calculate both sides: \(9 \times 9 = 81\) and \(8 \times 10 = 80\). Now compare the results: if 81 is less than or equal to 80, then the statement is true.

Step 4: Decide the Truth Value

Since 81 is greater than 80, the inequality \( 81 \leq 80 \) is false. Therefore, \( \frac{9}{10} \leq \frac{8}{9} \) is also false.

Key Concepts

Cross-Multiplication

Cross-multiplication is a simple yet powerful technique for comparing fractions without converting them to decimals. This method helps avoid errors that may occur during decimal conversion. To use cross-multiplication:

• Multiply the numerator of the first fraction with the denominator of the second fraction.
• Then, multiply the numerator of the second fraction with the denominator of the first fraction.

In our example with fractions \( \frac{9}{10} \) and \( \frac{8}{9} \):

• First, multiply 9 (numerator of the first fraction) by 9 (denominator of the second fraction) to get 81.
• Next, multiply 8 (numerator of the second fraction) by 10 (denominator of the first fraction) to get 80.

What you end up with are two simple products to compare, helping determine the size of each fraction quickly.

Inequalities

Inequalities are expressions used to compare values, showing that one value is larger or smaller than another. They use symbols such as \( \leq \), \( \geq \), \( < \), and \( > \). Understanding inequalities in the context of fractions is crucial for mathematical reasoning.
For the given inequality \( \frac{9}{10} \leq \frac{8}{9} \):

• It uses the "less than or equal to" symbol \( \leq \), suggesting that the first fraction is no greater than the second.
• The statement implies a comparison to check if one fraction is indeed less or perhaps equal to the other.

Keep in mind that effectively solving inequalities involving fractions often requires comparing their cross-multiplied products. This way, you can avoid mishaps with decimal approximations and ensure accurate comparisons.

Truth Value Determination

Truth value determination involves deciding whether a mathematical statement or inequality is true or false. This concept is fundamental in proving results in math. When it comes to fraction comparisons:

• First, establish whether the inequality is satisfied by checking the cross-multiplied results.
• In our problem: evaluate if \( 81 \leq 80 \).
• Since 81 is greater than 80, the statement \( \frac{9}{10} \leq \frac{8}{9} \) is false.

The truth evaluation is essential not only in understanding each fraction's relationship but also in laying the groundwork for more advanced mathematical concepts. It's a skill frequently used in algebra to solve equations and inequalities systematically. Always review the results for accuracy, especially in exams or real-life applications, to ensure your conclusions are correct.