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

Verified step by step guidance

1. Start by rewriting the complex fraction clearly: \(\frac{\frac{-2}{x+h} - \frac{-2}{x}}{h}\).
2. Simplify the numerator by combining the two fractions: \(\frac{-2}{x+h} - \left(-\frac{2}{x}\right) = \frac{-2}{x+h} + \frac{2}{x}\).
3. Find a common denominator for the numerator fractions, which is \(x(x+h)\), and rewrite each fraction with this common denominator: \(\frac{-2x}{x(x+h)} + \frac{2(x+h)}{x(x+h)}\).
4. Combine the fractions in the numerator over the common denominator: \(\frac{-2x + 2(x+h)}{x(x+h)}\).
5. Now, rewrite the entire expression as a division by \(h\): \(\frac{\frac{-2x + 2(x+h)}{x(x+h)}}{h} = \frac{-2x + 2(x+h)}{x(x+h)} \times \frac{1}{h}\).

Key Concepts

Complex Fractions

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

Finding Common Denominators

When subtracting or adding fractions, you must find a common denominator to combine them into a single fraction. This involves identifying the least common multiple of the denominators and rewriting each fraction accordingly.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms, factoring, and reducing fractions to their simplest form. This process helps to make expressions easier to work with and understand, especially when variables and multiple terms are involved.