Question

Perform the indicated operations and simplify the result. Leave your answer in factored form. $$ \frac{1+\frac{1}{x}}{1-\frac{1}{x}} $$

Short answer

\( \frac{x + 1}{x - 1} \)

Step-by-step solution

Simplify the Numerator

The numerator of the expression is \(1 + \frac{1}{x}\). To combine the terms, find a common denominator: \[1 + \frac{1}{x} = \frac{x}{x} + \frac{1}{x} = \frac{x + 1}{x}\]

Simplify the Denominator

The denominator of the expression is \(1 - \frac{1}{x}\). Similarly, find a common denominator: \[1 - \frac{1}{x} = \frac{x}{x} - \frac{1}{x} = \frac{x - 1}{x}\]

Combine the Simplified Numerator and Denominator

Now place the simplified numerator and denominator into the original expression: \[\frac{\frac{x + 1}{x}}{\frac{x - 1}{x}}\]

Simplify the Compound Fraction

To simplify the compound fraction, divide the numerator by the denominator: \[\frac{x + 1}{x} \div \frac{x - 1}{x} = \frac{x + 1}{x} \times \frac{x}{x - 1} = \frac{(x + 1)\cdot x}{x \cdot (x - 1)} = \frac{x + 1}{x - 1}\]

Final Answer in Factored Form

The fraction is already simplified and in factored form, so the final answer is: \[\frac{x + 1}{x - 1}\]

Key concepts

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This helps in making the fractions easier to understand and work with. To simplify a fraction, divide the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). For example, if we have the fraction \(\frac{6}{9}\), we find the GCD of 6 and 9, which is 3, then divide both the numerator and the denominator by 3: \(\frac{6 \/ 3}{9 \/ 3} = \frac{2}{3}\). This is easier to work with in subsequent calculations. Another important aspect of simplification is understanding when to stop. A fraction is considered simplified when the numerator and denominator are relatively prime, i.e., their only common divisor is 1.

Compound Fractions

A compound fraction (also known as a complex fraction) is a fraction where the numerator, the denominator, or both, contain fractions themselves. For example, an expression like \(\frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}\) is a compound fraction. These can look intimidating, but they become much simpler when broken down step by step. To simplify a compound fraction:

• First, simplify the fractions within the numerator and denominator.
• Next, rewrite the expression as a division of two fractions.
• Then, simplify by performing the division operation, which usually involves multiplying by the reciprocal of the denominator fraction.

In our exercise, we simplified the given compound fraction by finding common denominators within both the numerator and the denominator. This made it easier to rewrite the whole fraction in a simpler form and then performed the division.

Factored Form

Factored form refers to an expression rewritten as a product of its factors. This is particularly useful in algebra because it can simplify computations and provide clearer insight into the properties of the mathematical expressions. For instance, the fraction \( \frac{x+1}{x-1} \) is already in a factored form because numerator and denominator cannot be simplified further:

• Factoring helps in solving equations easily by setting each factor to zero (zero product property).
• It simplifies working with polynomials and rational expressions.

For our specific exercise, after simplifying the numerator \(\frac{x+1}{x}\) and the denominator \(\frac{x-1}{x}\) of our compound fraction, the final simplified result, \( \frac{x+1}{x-1} \), is left in factored form. This form tells us clearly that there are no common factors between numerator and denominator.