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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 75a
Chapter 1, Problem 75a

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

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Identify the complex fraction: \(\frac{1 + \frac{1}{1-b}}{1 - \frac{1}{1+b}}\).
Find a common denominator for the numerator: rewrite \(1\) as \(\frac{1-b}{1-b}\), so the numerator becomes \(\frac{1-b}{1-b} + \frac{1}{1-b} = \frac{(1-b) + 1}{1-b}\).
Simplify the numerator's numerator: \((1-b) + 1 = 2 - b\), so the numerator is \(\frac{2 - b}{1 - b}\).
Find a common denominator for the denominator: rewrite \(1\) as \(\frac{1+b}{1+b}\), so the denominator becomes \(\frac{1+b}{1+b} - \frac{1}{1+b} = \frac{(1+b) - 1}{1+b}\).
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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