Simplify each complex fraction. [ 1 + 1/(1-b) ] / [ 1 - 1/(1+b) ]

Verified step by step guidance

1. Identify the complex fraction: \(\frac{1 + \frac{1}{1-b}}{1 - \frac{1}{1+b}}\).
2. Find a common denominator for the numerator: rewrite \(1\) as \(\frac{1-b}{1-b}\), so the numerator becomes \(\frac{1-b}{1-b} + \frac{1}{1-b} = \frac{(1-b) + 1}{1-b}\).
3. Simplify the numerator's numerator: \((1-b) + 1 = 2 - b\), so the numerator is \(\frac{2 - b}{1 - b}\).
4. Find a common denominator for the denominator: rewrite \(1\) as \(\frac{1+b}{1+b}\), so the denominator becomes \(\frac{1+b}{1+b} - \frac{1}{1+b} = \frac{(1+b) - 1}{1+b}\).
5. Simplify the denominator's numerator: \((1+b) - 1 = b\), so the denominator is \(\frac{b}{1 + b}\). Now, rewrite the original complex fraction as \(\frac{\frac{2 - b}{1 - b}}{\frac{b}{1 + b}}\) and proceed by multiplying the numerator by the reciprocal of the denominator.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying complex fractions involves rewriting them as a single simple fraction by finding common denominators or multiplying numerator and denominator by the least common denominator.

Finding Common Denominators

To combine or simplify expressions with fractions, it is essential to find a common denominator. This allows you to add or subtract fractions by rewriting them with the same denominator, making the operations straightforward and enabling simplification.

Algebraic Simplification

Algebraic simplification involves combining like terms, factoring, and reducing expressions to their simplest form. In this problem, simplifying expressions with variables like 'b' requires careful manipulation to avoid errors and to express the final answer clearly.