Simplify each complex fraction. [ (x+4)/x - 3/(x-2) ] / [ x/(x - 2) + 1/x ]

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Identify the complex fraction: the numerator is \(\frac{x+4}{x} - \frac{3}{x-2}\) and the denominator is \(\frac{x}{x-2} + \frac{1}{x}\).

Find a common denominator for the numerator terms, which are \(\frac{x+4}{x}\) and \(\frac{3}{x-2}\). The common denominator is \(x(x-2)\).

Rewrite each term in the numerator with the common denominator \(x(x-2)\): multiply numerator and denominator accordingly to combine into a single fraction.

Similarly, find a common denominator for the denominator terms \(\frac{x}{x-2}\) and \(\frac{1}{x}\), which is also \(x(x-2)\), and combine them into a single fraction.

Now, divide the simplified numerator fraction by the simplified denominator fraction by multiplying the numerator fraction by the reciprocal of the denominator fraction, then simplify the resulting expression.

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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 involves rewriting the expression to eliminate the smaller fractions, often by finding a common denominator or multiplying numerator and denominator by the least common denominator (LCD).

Finding a Common Denominator

When subtracting or adding fractions, it is essential to find a common denominator to combine them into a single fraction. The common denominator is typically the least common multiple of the denominators involved, allowing the fractions to be expressed with the same base for straightforward addition or subtraction.

Simplifying Rational Expressions

Simplifying rational expressions involves factoring polynomials, canceling common factors, and reducing the expression to its simplest form. This process helps in making complex algebraic fractions easier to work with and is crucial when dealing with expressions that include variables in denominators.