Problem 34 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 3 or \(5 .\) $$ \frac{\frac{3}{x}-3}{\frac{9}{x}-3} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{1-x}{3-x} \).

Step 1: Identify the complex fraction

The fraction given is \( \frac{\frac{3}{x}-3}{\frac{9}{x}-3} \). It is complex because both the numerator and the denominator contain fractions.

Step 2: Create a common denominator in the numerator

In the numerator \( \frac{3}{x} - 3 \), convert \( 3 \) to have the same denominator: \( \frac{3}{x} - \frac{3x}{x} \).

Step 3: Simplify the numerator

Combine the terms in the numerator to get \( \frac{3-3x}{x} \).

Step 4: Create a common denominator in the denominator

In the denominator \( \frac{9}{x} - 3 \), convert \( 3 \) to have the same denominator: \( \frac{9}{x} - \frac{3x}{x} \).

Step 5: Simplify the denominator

Combine the terms in the denominator to get \( \frac{9-3x}{x} \).

Step 6: Simplify the overall fraction

The complex fraction becomes \( \frac{\frac{3-3x}{x}}{\frac{9-3x}{x}} \). Divide the terms: \( \frac{3-3x}{9-3x} \).

Step 7: Simplify the result fraction

Factor out 3 from both terms: \( \frac{3(1-x)}{3(3-x)} \). Cancel the common factor \( 3 \) from the numerator and denominator.

Step 8: Final Result

The simplified fraction is \( \frac{1-x}{3-x} \).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, the denominator, or both, contain fractions themselves. They may seem tricky at first, but they follow the same fundamental principles of fractions. The goal is to simplify them so they look less daunting. When you approach a complex fraction like \(\frac{\frac{3}{x}-3}{\frac{9}{x}-3}\), think of it as a problem within a problem.

• The first step is to identify it as a complex fraction because it involves operations within the numerator and denominator.
• The key to handling complex fractions is simplifying each part so you end up with a simple division problem.

After simplifying both the top and bottom parts separately, you then carry out the division to simplify it further.

Simplification

Simplification is the process of making an expression as straightforward as possible. With complex fractions, this involves reducing each part to a simpler form. When facing \(\frac{\frac{3}{x}-3}{\frac{9}{x}-3}\), it requires finding a common way to write the terms so they have a similar base. For simplification:

• Convert plain numbers into terms that can work with the fractions (e.g., converting \(3\) to \(\frac{3x}{x}\)).
• Combine like terms so you have a single fraction on top and another on the bottom.

Doing this means turning each part into a single simple fraction, making the overall fraction much simpler to handle.

Factoring

Factoring involves breaking down a number or expression into its multiplying components. It's like finding what you multiply together to get a certain expression. Consider the final simplification stage: \(\frac{3(1-x)}{3(3-x)}\).Here's how factoring helps in simplifying:

• Look for common factors in both the numerator and the denominator.
• When you find a common factor, like \(3\) in both parts in this example, you can cancel them out to reduce the fraction.

After factoring and canceling out, the complex fraction becomes much simpler: \(\frac{1-x}{3-x}\). This step is critical because it takes a more complex form and trims it down to its essential parts.

Common Denominator

Finding a common denominator is crucial when working with complex fractions. This process transforms the fractions into a compatible form, making them easier to combine or simplify. In the fractions \(\frac{3}{x}\) and \(3\), or \(\frac{9}{x}\) and \(3\), use the variable \(x\) as the common denominator.To create common denominators:

• Rewrite terms without a denominator to match the one they need (e.g., \(3\) becomes \(\frac{3x}{x}\)).
• Apply this method to both the numerator and the denominator of the complex fraction independently before simplifying.

By having a common denominator, it becomes much simpler to combine and then simplify the fractions. This step is foundational for simplifying both simple and complex fractions efficiently.