Problem 76 - Algebra A Combined Function

Question

Explain the similarities between subtracting \(\frac{3}{8}\) from \(\frac{7}{8}\) and subtracting \(\frac{6}{x+3}\) from \(\frac{9}{x+3}\).

Step-by-Step Solution

Answer

Both involve subtracting numerators of fractions with the same denominators.

Step 1: Identify the Denominators

First, observe the denominators in both subtraction problems. In the expression \(\frac{7}{8} - \frac{3}{8}\), the denominator is 8 for both fractions. Similarly, in the expression \(\frac{9}{x+3} - \frac{6}{x+3}\), the denominator is \(x + 3\) for both fractions.

Step 2: Subtract the Numerators

Since the denominators are the same in each case, you can subtract the numerators directly. For \(\frac{7}{8} - \frac{3}{8}\), the result is \(\frac{7-3}{8} = \frac{4}{8}\). For \(\frac{9}{x+3} - \frac{6}{x+3}\), the result is \(\frac{9-6}{x+3} = \frac{3}{x+3}\).

Step 3: Simplify the Resulting Fractions

In general, after performing the subtraction, the next step is to simplify the fractions if possible. \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\) by dividing the numerator and the denominator by 4. However, \(\frac{3}{x+3}\) cannot be simplified further because 3 and \(x+3\) have no common factors other than 1.

Step 4: Identify the Similarity

Notice that in both expressions, subtracting fractions with the same denominators allows for a straightforward subtraction of the numerators, which highlights a core principle of fraction operations: identical denominators enable direct operations with numerators.

Key Concepts

Common Denominators

When subtracting fractions, seeing the same denominator in both fractions is key. This feature, known as having a "common denominator," allows us to work on the numerators directly.
For example, in the fractions \( \frac{3}{8} \) and \( \frac{7}{8} \), the denominator is 8 for both. Similarly, the fractions \( \frac{6}{x+3} \) and \( \frac{9}{x+3} \) have a common denominator of \( x+3 \).
Having common denominators is crucial because it simplifies the process.

• Without common denominators, you'd first need to convert them to equivalent fractions with a common base.
• In our given examples, this step is already done, allowing us to focus directly on the numerators.

Recognizing a common denominator simplifies and accelerates the process of subtracting fractions.

Numerator Subtraction

Once we have common denominators, the next step is to subtract the numerators. This operation is straightforward thanks to their identical bases.
Consider the fractions \( \frac{7}{8} - \frac{3}{8} \). Since the denominator is already shared, subtract the numerators: \( 7 - 3 \). The result is \( \frac{4}{8} \).
Similarly, with \( \frac{9}{x+3} - \frac{6}{x+3} \), subtracting the numerators \( 9 - 6 \) gives us \( \frac{3}{x+3} \).

• The key point here is that numerator subtraction becomes simple and direct when the denominators match.
• This rule applies whether you are dealing with numbers, like 8, or algebraic expressions, like \( x+3 \).

Identical denominators allow us to ignore the denominators temporarily and focus solely on the numerators, making calculations quicker.

Simplifying Fractions

After subtracting the numerators, it's time to simplify the resulting fractions. Simplification makes fractions easier to read and compare.
In the case of our example, \( \frac{4}{8} \) can be simplified. We do this by finding the greatest common factor of 4 and 8. Both share a factor of 4, so divide them: \( \frac{4 \div 4}{8 \div 4} = \frac{1}{2} \).
On the other hand, for \( \frac{3}{x+3} \), no simplification occurs. The numerator is 3, and the denominator is \( x+3 \); they share no common factors other than 1.

• Simplifying fractions when possible helps in reducing complexity, making them more manageable, especially when dealing with larger expressions.
• However, if no simplification is possible, as in \( \frac{3}{x+3} \), the fraction remains as it is.

Simplification is an essential final step, ensuring our answer is clear and concise.