Problem 67 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\(\frac{(2-b)(1+b)}{(1-b)b}\) is the simplified form.

Step 1: Simplifying the Numerator

Start by simplifying the numerator of the complex fraction: \(1 + \frac{1}{1-b}\). To do this, we will get a common denominator for the terms. The common denominator is \(1-b\). This allows us to write: \[1 + \frac{1}{1-b} = \frac{1(1-b)}{1-b} + \frac{1}{1-b} = \frac{1-b+1}{1-b} = \frac{2-b}{1-b}\]

Step 2: Simplifying the Denominator

Next, simplify the denominator: \(1 - \frac{1}{1+b}\). Similarly, we need a common denominator, which is \(1+b\). This gives:\[1 - \frac{1}{1+b} = \frac{1(1+b)}{1+b} - \frac{1}{1+b} = \frac{1+b-1}{1+b} = \frac{b}{1+b}\]

Step 3: Division of Two Fractions

Now simplify the original complex fraction by dividing the simplified numerator by the simplified denominator:\[\frac{\frac{2-b}{1-b}}{\frac{b}{1+b}} = \frac{2-b}{1-b} \times \frac{1+b}{b}\]

Step 4: Multiplying Fractions

Multiply the fractions from Step 3 by multiplying the numerators together and the denominators together:\[\frac{(2-b)(1+b)}{(1-b)(b)}\]

Step 5: Final Simplification

Perform any possible simplifications on the result from Step 4. Since \(2-b\) and \(1-b\) cannot be further simplified together, and \(1+b\) and \(b\) have no common factors, the expression:\[\frac{(2-b)(1+b)}{(1-b)(b)}\] is already simplified.

Key Concepts

Numerator and Denominator Simplification

Breaking down complex fractions can be intimidating, but breaking them into simpler parts can make them much easier to handle. The first step is often to work on each fraction separately within the complex fraction. Let's begin with the numerator:

• Consider the numerator: \(1 + \frac{1}{1-b}\). To simplify, find a common denominator. Here, it is \(1-b\). This means you rewrite \(1\) as \(\frac{1(1-b)}{1-b}\), making the common denominator visible.
• Combine the terms: \(\frac{1(1-b)}{1-b} + \frac{1}{1-b} = \frac{1-b+1}{1-b} = \frac{2-b}{1-b}\).

Next, handle the denominator of the complex fraction by applying the same principle:

• Analyze \(1 - \frac{1}{1+b}\). Again, use a common denominator, \(1+b\).
• Rewrite \(1\) to match this common denominator: \(\frac{1(1+b)}{1+b}\).
• Merge the terms: \(\frac{1+b-1}{1+b} = \frac{b}{1+b}\).

By simplifying both the numerator and the denominator, you bring clarity and reduce the complexity of the original problem, setting a solid stage for the subsequent operations.

Fraction Division

When you have a fraction divided by another fraction, there is a straightforward procedure to follow. Fraction division can be simplified by the 'invert and multiply' rule.

• The expression \(\frac{\frac{2-b}{1-b}}{\frac{b}{1+b}}\) can be seen as a division of two fractions.
• According to the rule, instead of dividing by a fraction, you multiply by its reciprocal. This turns the operation into multiplication: \(\frac{2-b}{1-b} \times \frac{1+b}{b}\).

Learning this step is crucial as it sets up the stage for combining fractions in a more manageable form. It strips away the complex division aspect and transitions into something more familiar and fundamentally simple, multiplication of fractions.

Multiplying Fractions

Once you have converted the division into multiplication by taking the reciprocal, multiplying fractions is all about handling the numerators and denominators separately:

• Take the numerators: \((2-b)\) and \((1+b)\), and multiply them to get \((2-b)(1+b)\).
• Do the same for the denominators: \((1-b)\) and \(b\), resulting in \((1-b)(b)\).

Thus, the product of these fractions becomes:\[\frac{(2-b)(1+b)}{(1-b)(b)}\]This expression is a result of directly combining the numerators and denominators from the fractions involved. Remember, always look for potential simplifications thereafter. However, in this case, no common factors exist between the terms in the numerator and the denominator, confirming it is already in its simplest form.