Problem 11 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{3}{4 x}+\frac{8}{x-2}\)

Step-by-Step Solution

Answer

The solution is \( \frac{35x - 6}{4x(x-2)} \).

Step 1: Identifying the Problem

The exercise requires us to add two algebraic fractions: \( \frac{3}{4x} \) and \( \frac{8}{x-2} \). To add these fractions, we need a common denominator.

Step 2: Finding a Common Denominator

The denominators are \( 4x \) and \( x-2 \). Since these are distinct terms, the common denominator will be the product of both denominators: \( 4x(x-2) \).

Step 3: Rewriting Each Fraction

We rewrite each fraction with the common denominator. For \( \frac{3}{4x} \), multiply by \( \frac{x-2}{x-2} \) to get \( \frac{3(x-2)}{4x(x-2)} \). For \( \frac{8}{x-2} \), multiply by \( \frac{4x}{4x} \) to get \( \frac{8 \times 4x}{4x(x-2)} = \frac{32x}{4x(x-2)} \).

Step 4: Adding the Fractions

Now that both fractions have the common denominator \( 4x(x-2) \), add the numerators: \( \frac{3(x-2) + 32x}{4x(x-2)} \).

Step 5: Simplifying the Expression

Expand and combine like terms in the numerator: \( 3(x-2) = 3x - 6 \), so the expression becomes \( \frac{3x - 6 + 32x}{4x(x-2)} = \frac{35x - 6}{4x(x-2)} \).

Step 6: Final Answer

The fraction \( \frac{35x - 6}{4x(x-2)} \) cannot be simplified further as there are no common factors to cancel.

Key Concepts

Common Denominator

When dealing with algebraic fractions, finding a common denominator is crucial for operations like addition or subtraction. The common denominator, much like in numerical fractions, is a value that both denominators can divide into without leaving a remainder. This helps to combine the fractions seamlessly.
To find a common denominator for algebraic fractions such as \( \frac{3}{4x} \) and \( \frac{8}{x-2} \), we look at each denominator separately. Since \( 4x \) and \( x-2 \) are distinct, neither can be converted directly into the other through multiplication by a constant.
Thus, the common denominator for these fractions becomes the product of the two denominators: \( 4x(x-2) \). This approach ensures that both fractions are transformed without altering their values, and allows for straightforward addition or subtraction.

• Example: For \( \frac{1}{2} \) and \( \frac{1}{3} \), a common denominator is \( 6 \) because \( 2 \times 3 = 6 \).
• Similarly, for algebraic terms \( 4x \) and \( x-2 \), \( 4x(x-2) \) serves effectively.

This strategy simplifies the process of manipulating fractions and paves the way for the next step: rewriting and adding the fractions.

Simplifying Fractions

Once fractions are rewritten with a common denominator, it's crucial to simplify them whenever possible to make the expression easier to work with. Simplifying involves reducing the fraction to its simplest form.
After combining the numerators, as we did with \( \frac{3(x-2) + 32x}{4x(x-2)} \), it results in a single fraction, \( \frac{35x - 6}{4x(x-2)} \). It's essential to check both the numerator and the denominator for any common factors that might simplify further.
Begin by expanding terms in the numerator if needed, such as expanding \( 3(x-2) \) to \( 3x - 6 \). After merging it with \( 32x \), we have the simpler form \( 35x - 6 \).

• If there were common factors, like a number or variable, present in both the numerator and denominator, we could simplify by canceling them out.
• In this specific case, we can't simplify further as there are no shared factors in the final expression.

Ensuring a fraction is fully simplified is a key step in managing algebraic expressions neatly and is often a requirement in correct mathematical solutions.

Adding Fractions

Adding fractions, particularly algebraic ones, follows a process similar to numerical fractions. Once fractions share a common denominator, they can be added directly by summing their numerators.
For algebraic fractions like \( \frac{3}{4x} \) and \( \frac{8}{x-2} \), rewriting them as \( \frac{3(x-2)}{4x(x-2)} \) and \( \frac{32x}{4x(x-2)} \) gives the same denominator \( 4x(x-2) \). This allows their numerators to be combined seamlessly into one expression: \( 3(x-2) + 32x \).
This step focuses only on the numerators, creating \( \frac{3x - 6 + 32x}{4x(x-2)} \), which simplifies the operation while maintaining the overall value of the fractions.

• The denominators stay the same because combining fractions relies on a shared foundation.
• By adding numerators directly, only one operation - simplification - is needed after summing.

This method helps to streamline solving complex equations and keeps calculations precise, ensuring their accuracy throughout the solution process.