Problem 45 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Example 6 $$ \frac{\frac{1}{x+1}}{1+\frac{1}{x+1}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{1}{x+2} \).

Step 1: Identify the Components

The complex fraction is \( \frac{\frac{1}{x+1}}{1+\frac{1}{x+1}} \). Identify that the numerator is \( \frac{1}{x+1} \) and the denominator is \( 1 + \frac{1}{x+1} \).

Step 2: Simplify the Denominator

To simplify the denominator \( 1 + \frac{1}{x+1} \), find a common denominator, which is \( x+1 \). Thus, rewrite 1 as \( \frac{x+1}{x+1} \), so the expression becomes \( \frac{x+1}{x+1} + \frac{1}{x+1} = \frac{x+1+1}{x+1} = \frac{x+2}{x+1} \).

Step 3: Rewrite the Complex Fraction

Replace the denominator in the original complex fraction with the simplified version: \( \frac{\frac{1}{x+1}}{\frac{x+2}{x+1}} \). This expression is now division of two fractions.

Step 4: Divide the Fractions

To divide fractions, multiply by the reciprocal of the divisor. Therefore, \( \frac{\frac{1}{x+1}}{\frac{x+2}{x+1}} \) becomes \( \frac{1}{x+1} \times \frac{x+1}{x+2} \).

Step 5: Simplify the Expression

The \( x+1 \) terms in the numerator and the denominator cancel each other, simplifying the expression to \( \frac{1}{x+2} \).

Key Concepts

Simplification of Complex Fractions

When simplifying complex fractions, the goal is to reduce the expression into a simpler form. This process often involves several key steps:

• Identifying the components. Recognize and separate the numerator and the denominator.
• Simplifying components. This includes finding a common denominator if required.
• Rewriting the expression by using these simplified components.

A complex fraction is essentially a fraction with fractions in its numerator, its denominator, or both. In the original exercise, you begin by identifying the parts to simplify. The numerator is already a simple fraction: \( \frac{1}{x+1} \). The denominator, however, \( 1 + \frac{1}{x+1} \), needs simplification.
To do so, rewrite 1 with the common denominator \( x+1 \), as \( \frac{x+1}{x+1} \), which allows the expression to become \( \frac{x+2}{x+1} \). After simplifying both parts, the complex fraction looks less complex.

Division of Fractions

When dealing with fraction division, the standard rule is to multiply by the reciprocal.

• Fraction Division Rule: Multiply the first fraction by the reciprocal of the second.
• Reciprocal: The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).

In the complex fraction \( \frac{\frac{1}{x+1}}{\frac{x+2}{x+1}} \), you apply this rule by flipping the second fraction (the divisor).
This means, you transform the division into a multiplication: \( \frac{1}{x+1} \times \frac{x+1}{x+2} \). The simplification of multiplying gives us one step closer to a simpler form.

Understanding Reciprocals

The reciprocal is a fundamental concept in mathematics, especially useful in division operations. A reciprocal of a number or a fraction is what gives 1 when multiplied with the given number.

• For a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
• Multiplying a fraction by its reciprocal always results in 1.
• Reciprocals simplify fraction division into manageable multiplication.

Returning to the earlier problem, by taking the reciprocal of the divisor \( \frac{x+2}{x+1} \) and multiplying, we effectively divide by the given fraction. This use of the reciprocal transforms a complex process into a straightforward multiplication, facilitating the final simplification to \( \frac{1}{x+2} \).
Understanding reciprocals not only aids in simplifying complex fractions but is also a critical skill across various mathematics topics.