Simplify each complex fraction. [ 2 + 2/(1+x) ] / [ 2 - 2/(1-x) ]

Verified step by step guidance

1. Identify the complex fraction: \(\frac{2 + \frac{2}{1+x}}{2 - \frac{2}{1-x}}\).
2. Find a common denominator for the numerator expression \(2 + \frac{2}{1+x}\). Rewrite \(2\) as \(\frac{2(1+x)}{1+x}\) to combine the terms: \(\frac{2(1+x)}{1+x} + \frac{2}{1+x}\).
3. Combine the numerator terms over the common denominator \$1+x$: \(\frac{2(1+x) + 2}{1+x}\).
4. Similarly, find a common denominator for the denominator expression \(2 - \frac{2}{1-x}\). Rewrite \(2\) as \(\frac{2(1-x)}{1-x}\) and combine: \(\frac{2(1-x)}{1-x} - \frac{2}{1-x} = \frac{2(1-x) - 2}{1-x}\).
5. Rewrite the original complex fraction as a division of two fractions: \(\frac{\frac{2(1+x) + 2}{1+x}}{\frac{2(1-x) - 2}{1-x}}\). Then multiply by the reciprocal of the denominator fraction: \(\frac{2(1+x) + 2}{1+x} \times \frac{1-x}{2(1-x) - 2}\).

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 terms with different denominators, identify the least common denominator (LCD). This allows you to rewrite each term with the same denominator, enabling addition or subtraction of fractions within the numerator or denominator of the complex fraction.

Simplifying Rational Expressions

After combining fractions, simplify the resulting rational expression by factoring and canceling common factors. This step reduces the expression to its simplest form, making it easier to interpret or use in further calculations.