Problem 63 - A Graphical Approach to College Algebra

Question

Simplify each complex fraction. $$\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$$

Step-by-Step Solution

Answer

The complex fraction simplifies to \( \frac{x+1}{x-1} \).

Step 1: Analyze the Complex Fraction

The given complex fraction is \( \frac{1+\frac{1}{x}}{1-\frac{1}{x}} \). Notice that the numerator and the denominator are both expressions involving the fraction \( \frac{1}{x} \).

Step 2: Rewrite with a Common Denominator

To simplify, rewrite both the numerator and the denominator to combine terms. The numerator \( 1+\frac{1}{x} \) can be written as \( \frac{x}{x} + \frac{1}{x} = \frac{x+1}{x} \). Similarly, the denominator \( 1-\frac{1}{x} \) becomes \( \frac{x}{x} - \frac{1}{x} = \frac{x-1}{x} \).

Step 3: Express as a Single Fraction

The complex fraction now looks like this: \[ \frac{\frac{x+1}{x}}{\frac{x-1}{x}} \].When dividing by a fraction, multiply by its reciprocal. Therefore, this can be rewritten as:\[ \frac{x+1}{x} \times \frac{x}{x-1} \].

Step 4: Simplify the Multiplication

The expression \( \frac{x+1}{x} \times \frac{x}{x-1} \) simplifies by cancelling the \( x \) in the numerator of the first fraction with the \( x \) in the denominator of the second fraction, resulting in \( \frac{x+1}{x-1} \). This is the simplified form.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra, making complex fraction calculations easier and more manageable. To simplify a fraction, you first need to find the greatest common factor (GCF) of the numerator and the denominator. This helps in reducing a fraction to its simplest form.

In the case of complex fractions, which contain a fraction in either the numerator, denominator, or both, the process involves a few more steps.

• First, identify and simplify the smaller fractions in the numerator and denominator.
• Next, seek a common denominator to combine terms within the numerator and the denominator.
• Then, convert the complex fraction into a simple division problem.
• Simplify further to ensure the fraction is in its simplest form.

By following these steps, you transform a seemingly complex problem into something more straightforward and manageable.

Algebraic Expressions

An algebraic expression is a blend of numbers, variables, and operations. These expressions can be simple, involving just one variable and a few operations, or complex, involving multiple variables and operations.

In algebra, you often encounter expressions like \( x + 2 \) or \( 3x - 4 \), but complex fractions are slightly more involved. They have fractions within a larger fraction, which can make them look intimidating at first glance.

When dealing with algebraic expressions in complex fractions, it's important to:

• Recognize that each part of the expression follows algebraic rules.
• Combine like terms whenever possible.
• Use factoring techniques to simplify the expression.

This structured approach allows you to manipulate and simplify algebraic expressions within fractions effectively.

Fractions with Variables

Fractions with variables can add a layer of complexity to algebraic problems, primarily because both numbers and letters now play a role in the calculation. Understanding how to work with them is crucial for mastering algebra.

To simplify fractions that include variables, also known as rational expressions, you need to:*

• Identify and state any restrictions on the variable, such as values that make the denominator zero.
• Factor both the numerator and the denominator if possible.
• Cancel any common factors.
• Be mindful of the domain and ensure the operations maintain the integrity of the expression.

By being systematic and precise, you maintain control over the variables within the problem, allowing for correct simplification and interpretation of expressions with variables.