Problem 36 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{25}{x+5}+5}{\frac{3}{x+5}-5} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{5(x + 10)}{-5x - 22} \).

Step 1: Identify the Complex Fraction

The complex fraction given is \( \frac{\frac{25}{x+5}+5}{\frac{3}{x+5}-5} \). It consists of a fraction in the numerator \( \frac{25}{x+5} + 5 \) and a fraction in the denominator \( \frac{3}{x+5} - 5 \).

Step 2: Simplify the Numerator

To simplify the numerator \( \frac{25}{x+5} + 5 \), find a common denominator, which is \( x+5 \). Rewrite 5 as \( \frac{5(x+5)}{x+5} = \frac{5x+25}{x+5} \). Thus, the numerator becomes: \[ \frac{25}{x+5} + \frac{5x+25}{x+5} = \frac{25 + 5x + 25}{x+5} = \frac{5x + 50}{x+5} \].

Step 3: Simplify the Denominator

For the denominator \( \frac{3}{x+5} - 5 \), rewrite 5 as \( \frac{5(x+5)}{x+5} = \frac{5x+25}{x+5} \). Then the denominator becomes: \[ \frac{3}{x+5} - \frac{5x+25}{x+5} = \frac{3 - 5x - 25}{x+5} = \frac{-5x - 22}{x+5} \].

Step 4: Divide the Numerator by the Denominator

To simplify the complex fraction, divide the simplified numerator \( \frac{5x + 50}{x+5} \) by the simplified denominator \( \frac{-5x - 22}{x+5} \). This is equivalent to multiplying the numerator by the reciprocal of the denominator:\[ \frac{5x + 50}{x+5} \times \frac{x+5}{-5x - 22} = \frac{5x + 50}{-5x - 22} \].

Step 5: Simplify Further if Possible

Check if the fractions can be simplified further. Here, 5 can be factored out from the numerator:\[ \frac{5(x + 10)}{-5x - 22} \]. However, the numerator and denominator don't have common factors, so this is the final simplified form.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is a key part of solving complex fractions. An algebraic expression is a combination of numbers, variables, and arithmetic operations. For example, in our problem, expressions such as \( \frac{25}{x+5} + 5 \) and \( \frac{3}{x+5} - 5 \) are algebraic expressions. These include components like:\

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• **Variables**: Represent unknown values, like \( x \) in this exercise.
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• **Numbers**: Known values in the expression, such as 25 or 3.
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• **Operators**: Arithmetic symbols (e.g., +, -, ×, ÷) that indicate the operations to be performed.
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\Recognizing these components helps in identifying structural patterns and simplifying. Look for opportunities to factor or find a common denominator, as this simplifies further operations.

Numerators and Denominators

Numerators and denominators are critical when dealing with fractions, especially complex ones. In a fraction \( \frac{a}{b} \), \( a \) is the numerator and \( b \) is the denominator. Understanding them allows us to transform and simplify fractions accurately.\
\In the original exercise, the numerator is \( \frac{25}{x+5} + 5 \), and the denominator is \( \frac{3}{x+5} - 5 \). Simplifying these expressions involves adding or subtracting fractions. To do this,\

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• **Find a common denominator**: This helps combine terms in the numerator or denominator.
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• **Rewriting fractions**: Convert whole numbers like 5 into a fraction with a denominator of \( x+5 \) to allow for addition or subtraction.
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\By managing numerators and denominators effectively, you can combine expressions and reduce their complexity.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form. For complex fractions, this often means turning division into multiplication. You do this by multiplying the numerator by the reciprocal of the denominator. Here's a simplified process to follow:\
\1. **Identify and simplify components**: Simplify both the numerator and denominator separately.\2. **Divide using the reciprocal**: Convert the division of the fractions to multiplication by flipping the denominator and multiplying.\3. **Factor and reduce**: Factor possible components and cancel out any common terms in the numerator and denominator.\
\For example, in our case, simplify \( \frac{5x + 50}{x+5} \) and \( \frac{-5x - 22}{x+5} \) separately before multiplying the numerator by the reciprocal of the denominator. This approach aids in achieving a cleaner and more understandable fraction. By practicing these steps, you’ll master the art of fraction simplification!