Problem 20 - 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.

Step 1: Understand Inequality Symbols

The inequality symbol \(\leq\) means "less than or equal to." Thus, we need to determine whether \(\frac{9}{10}\) is less than or equal to \(\frac{8}{9}\).

Step 2: Find a Common Denominator

To compare two fractions, it's often helpful to find a common denominator. In this case, the denominators are 10 and 9. The least common multiple of 10 and 9 is 90.

Step 3: Convert Fractions to Equivalent Fractions

Convert \(\frac{9}{10}\) and \(\frac{8}{9}\) to equivalent fractions with a denominator of 90. Multiply the numerator and denominator of \(\frac{9}{10}\) by 9 to get \(\frac{81}{90}\). Multiply the numerator and denominator of \(\frac{8}{9}\) by 10 to get \(\frac{80}{90}\).

Step 4: Compare the New Fractions

Now compare \(\frac{81}{90}\) and \(\frac{80}{90}\). Since 81 is greater than 80, \(\frac{81}{90}\) is greater than \(\frac{80}{90}\).

Step 5: Conclude the Inequality

Since \(\frac{81}{90} > \frac{80}{90}\), this means \(\frac{9}{10}\) is greater than \(\frac{8}{9}\). Thus, the statement \(\frac{9}{10} \leq \frac{8}{9}\) is false.

Key Concepts

Comparing Fractions

When comparing fractions, we aim to determine which fraction is larger or if they might be equal. Fractions consist of two parts: a numerator (top part) and a denominator (bottom part). The key is understanding how these work together to show a portion of a whole. Consider two scenarios: when fractions have the same denominator and when they don't.

• If fractions have the same denominator, you can compare them by simply looking at the numerators. For example, with \( \frac{3}{5} \) and \( \frac{4}{5} \), you compare 3 and 4 directly. Since 4 is greater than 3, \( \frac{4}{5} \) is larger.
• If the denominators differ, a common technique is to find equivalent fractions with a common denominator. This way, you make a fair comparison since both fractions represent the same divisions of a whole.

Understanding the concept of comparing fractions is crucial for solving inequalities involving fractions, as it involves determining which value is less than, greater than, or equivalent to another.

Common Denominator

To effectively compare fractions with different denominators, we find a common denominator. A common denominator allows the fractions to be expressed in parts of the same size, making comparison straightforward.

• Least Common Multiple (LCM): The most efficient common denominator is the least common multiple of the denominators of the fractions. For denominators 10 and 9, the LCM is 90.
• Converting Fractions: Once you have the common denominator, convert each fraction to an equivalent one sharing this new denominator. Multiply the numerator and denominator of each original fraction by the necessary numbers to align them with the common denominator.

Using a common denominator helps you to directly compare the sizes of two fractions with different denominators, making it clear which one is larger or smaller.

Equivalent Fractions

Equivalent fractions may look different, but they actually represent the same portion of the whole. They are a crucial tool when comparing or adding fractions.

• Creating Equivalent Fractions: Multiply or divide both the numerator and the denominator by the same non-zero number. Doing this doesn't change the value of the fraction, only its appearance. For example, \( \frac{9}{10} \) can become \( \frac{81}{90} \) by multiplying both the numerator and denominator by 9.
• Uses in Comparison: By converting fractions to equivalent forms with common denominators, you simplify the process of comparison. You maintain the integrity of the original values, making it easy to see which fraction is larger or smaller.

Learning to create and recognize equivalent fractions is essential in understanding and solving fraction-related inequalities and other mathematical problems.