Problem 76 - Elementary Algebra

Question

For Problems \(73-76\), simplify each complex fraction. $$ \frac{5 x}{3+\frac{1}{x}}+2 $$

Step-by-Step Solution

Answer

\( \frac{5x^2 + 6x + 2}{3x+1} \)

Step 1: Identify the Complex Fraction

The expression given is \( \frac{5x}{3 + \frac{1}{x}} + 2 \). The complex fraction is \( \frac{5x}{3 + \frac{1}{x}} \), which means we need to simplify the fraction in the denominator first, \( 3 + \frac{1}{x} \).

Step 2: Simplify the Denominator

To simplify \( 3 + \frac{1}{x} \), we'll find a common denominator for the terms. The common denominator is \( x \). Express \( 3 \) as \( \frac{3x}{x} \). So, \( 3 + \frac{1}{x} = \frac{3x}{x} + \frac{1}{x} = \frac{3x + 1}{x} \).

Step 3: Rewrite the Complex Fraction

Replace the original denominator in the fraction with the simplified form: \( \frac{5x}{\frac{3x + 1}{x}} + 2 \).

Step 4: Simplify the Complex Fraction

When dividing by a fraction, multiply by its reciprocal. So, \( \frac{5x}{\frac{3x+1}{x}} = 5x \cdot \frac{x}{3x+1} = \frac{5x^2}{3x+1} \).

Step 5: Complete the Expression

Once the complex fraction is simplified, add 2 to the result: \( \frac{5x^2}{3x+1} + 2 \). To combine into a single fraction, rewrite 2 as \( \frac{2(3x+1)}{3x+1} = \frac{6x + 2}{3x+1} \). Adding these gives: \( \frac{5x^2 + 6x + 2}{3x+1} \).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They make up the formulas and equations you encounter in algebra. When dealing with algebraic expressions in problems, it's essential to understand the components:

• Variables: These are symbols, such as \( x \), which represent unknown values.
• Constants: These are fixed values like numbers, for example, \( 3 \) or \( 5 \).
• Operators: Symbols that denote mathematical operations like addition \((+)\), subtraction \((-\)), multiplication \((\cdot)\), and division \((\div)\).

In the expression given, \( \frac{5x}{3+\frac{1}{x}} + 2 \), \( 5x \) is a term involving multiplication of a constant and a variable, while \( 3 + \frac{1}{x} \) comprises addition and division. Being able to parse these expressions is crucial for solving complex fractions effectively.

Simplifying Fractions

Simplifying fractions means expressing a fraction in its simplest form. This often involves reducing the numerator and the denominator to their simplest terms by dividing them by their greatest common factor (GCF). For complex fractions, which involve fractions in the numerator, denominator, or both, this process is slightly more involved because it may require additional steps:

• First, simplify any algebraic expressions within the numerator or denominator.
• Next, find a way to eliminate smaller fractions by finding a common denominator or multiplying by the reciprocal.

The aim is to reduce the complex fraction to a simple fraction that no longer contains fractions within it. This was accomplished in the problem by eliminating the fraction within the denominator and rewriting it in a simpler form, eventually allowing multiplication by the reciprocal.

Common Denominators

Finding a common denominator is a key step when simplifying complex fractions. A common denominator is a shared multiple of the denominators in fractions, allowing us to combine them into a single expression. In the context of our problem with the expression \( 3 + \frac{1}{x} \):

• The common denominator for the terms \( 3 \) and \( \frac{1}{x} \) is \( x \).
• By expressing \( 3 \) as \( \frac{3x}{x} \), both terms in the denominator share \( x \) as the common denominator.
• This allows us to rewrite the denominator \( 3 + \frac{1}{x} \) as a single fraction \( \frac{3x + 1}{x} \).

Using a common denominator streamlines fractions and is essential when working to simplify them into a more manageable form.

Reciprocal Multiplication

Reciprocal multiplication is a technique used to simplify division by fractions. When dividing by a fraction, you multiply by its reciprocal. This is crucial in working with complex fractions:

• The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). By flipping the numerator and denominator, we convert division into multiplication.
• Within the exercise, we took \( \frac{5x}{\frac{3x+1}{x}} \) and transformed it by multiplying by the reciprocal of the denominator. So, \( 5x \) became multiplied by \( \frac{x}{3x+1} \).
• This step effectively reduces the complex fraction to \( \frac{5x^2}{3x+1} \).

Understanding and applying reciprocal multiplication helps tackle the challenge of simplifying complex fractions, converting problems into easier multiplication tasks.