Problem 35 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{8}{x+4}+2}{\frac{12}{x+4}-2} $$

Step-by-Step Solution

Answer

The complex fraction simplifies to \(-\frac{x+8}{x-2}\).

Step 1: Identify the Structure

The given expression is a complex fraction of the form \(\frac{\frac{8}{x+4}+2}{\frac{12}{x+4}-2}\). It is composed of fractions in both the numerator and the denominator. The first task is to combine the terms in the numerator and the denominator separately.

Step 2: Combine the Numerator

The numerator is \(\frac{8}{x+4} + 2\). To combine these, convert 2 into a fraction with the same denominator:\(\frac{2(x+4)}{x+4} = \frac{2x+8}{x+4}\). Now add: \[\frac{8}{x+4} + \frac{2x + 8}{x+4} = \frac{8 + 2x + 8}{x+4} = \frac{2x + 16}{x+4}.\]

Step 3: Combine the Denominator

The denominator is \(\frac{12}{x+4} - 2\). Convert 2 into a fraction with the same denominator: \(\frac{2(x+4)}{x+4} = \frac{2x+8}{x+4}\). Now subtract: \[\frac{12}{x+4} - \frac{2x + 8}{x+4} = \frac{12 - 2x - 8}{x+4} = \frac{-2x + 4}{x+4}.\]

Step 4: Divide and Simplify

The simplified expression is now \(\frac{\frac{2x+16}{x+4}}{\frac{-2x+4}{x+4}}\). To divide these fractions, multiply the first fraction by the reciprocal of the second: \[\frac{2x+16}{x+4} \times \frac{x+4}{-2x+4} = \frac{2x+16}{-2x+4}.\] Cancel the \(x+4\) terms in the numerator and denominator. Lastly, factor and simplify the resulting expression: \[\frac{2(x+8)}{-2(x-2)} = -\frac{x+8}{x-2}.\]

Step 5: Final Simplification

The simplified expression is \(-\frac{x+8}{x-2}\). Since no more common factors exist, this is the final simplified form.

Key Concepts

Simplification

Simplification, especially in complex fractions, revolves around making the expression easier to understand and use. A complex fraction typically has a fraction in its numerator, denominator, or both. Our goal is to transform this into a much simpler single fraction.
The first step involves managing and simplifying separate components of the complex fraction. In this case, we start by combining like terms in the numerators and denominators separately.

• Change constant terms into fractions with common denominators.
• Add or subtract the fractions as you would normally do, but keep the results in fractional form.

Using these strategies, we gradually condense a complicated fraction into a simpler form. The final step is often to cancel out any common factors shared by the numerator and the denominator.

Numerator and Denominator

The numerator and denominator are the building blocks of any fraction. In complex fractions, understanding and manipulating these parts is key to simplification.
Both the numerator and the denominator may contain fractions themselves, as seen in the given exercise:

• The numerator: \(\frac{8}{x+4} + 2\)
• The denominator: \(\frac{12}{x+4} - 2\)

Begin by rewriting the whole numbers as fractions with a common denominator. This step converts expressions to make subtraction or addition straightforward:
- Rewrite 2 as \(\frac{2(x+4)}{x+4}\) in both the numerator and the denominator.
Once you have common denominators, add or subtract the fractions accordingly, leading towards a new fraction for both parts of the complex expression.

Fraction Operations

Operations with fractions—such as addition, subtraction, multiplication, and division—are central to managing complex fractions effectively.
When working with complex fractions, remember:

• To simplify a fraction division, multiply by the reciprocal of the divisor.
• Combine fractions by finding a common denominator for addition or subtraction.

For example, once the numerators and denominators in this exercise were combined, the complex fraction was simplified further by turning the division into multiplication by the reciprocal.
This critical step ensures you can consolidate the entire expression into a simpler form, thereby concluding the problem with a clean solution.