Problem 46 - 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 complex fraction is \( \frac{1}{x-2} \).

Step 1: Identify the Numerator and Denominator

The given expression is a complex fraction. It has a fraction in its numerator and another fraction within its denominator. Let's identify them: the numerator is \( \frac{1}{x-1} \) and the denominator is \( 1 - \frac{1}{x-1} \).

Step 2: Simplify the Denominator

To simplify \( 1 - \frac{1}{x-1} \), find a common denominator. Rewrite 1 as \( \frac{x-1}{x-1} \) to get the common base: \( \frac{x-1}{x-1} - \frac{1}{x-1} = \frac{x-1-1}{x-1} = \frac{x-2}{x-1} \).

Step 3: Write the Complex Fraction

Now that the denominator is simplified, rewrite the entire complex fraction as: \[ \frac{\frac{1}{x-1}}{\frac{x-2}{x-1}} \].

Step 4: Simplify the Complex Fraction

To simplify this, multiply the numerator by the reciprocal of the denominator: \[ \frac{1}{x-1} \times \frac{x-1}{x-2} \].

Step 5: Cancel Common Factors

The \( x-1 \) in the numerator and denominator cancels out, yielding: \[ \frac{1}{x-2} \].

Key Concepts

Simplifying Fractions

Simplifying fractions is a key skill in algebra. It allows you to transform complex expressions into manageable forms. To simplify a fraction, you want to make the numerator and denominator as simple as possible.

Imagine you have a complicated fraction like our original problem. It can have fractions within fractions, known as a complex fraction. Simplifying involves several steps:

• Identify the fraction's parts: this means clearly recognizing both the numerator and the denominator.
• Rework each part to its simplest form.
• Combine everything back into a single, simple fraction.

By simplifying, not only do you make calculations easier but you also gain a clearer understanding of the expression's meaning.

Numerator and Denominator

Understanding the numerator and denominator is crucial because they define fractions. The numerator is the top part of the fraction, stating how many parts you have. The denominator is the bottom part, showing how many parts the whole is divided into.

In the complex fraction tackled here, our numerator is \(\frac{1}{x-1}\), and the denominator is \(1 - \frac{1}{x-1}\).

• To manage these parts effectively, simplify each separately when they themselves contain fractions.
• Finding a common denominator helps in rearranging terms and further simplifying.

Splitting a fraction into its components can frequently make the process of simplifying much more straightforward.

Reciprocal

The concept of a reciprocal is fundamental in dividing fractions. A reciprocal basically "flips" the fraction, making the numerator the denominator and vice versa.

This principle is especially useful when dealing with complex fractions. In our simplified complex fraction, once you've rearranged and simplified the denominator, the next step is key:

• Convert the denominator to its reciprocal.
• Multiply this reciprocal by the numerator.

This method allows you to simplify the original complex fraction into a much more understandable form, as seen when the numerator \(\frac{1}{x-1}\) multiplies the reciprocal of the denominator \(\frac{x-1}{x-2}\), resulting in \(\frac{1}{x-2}\). Understanding and using reciprocals makes the process of fraction division and simplification efficient.