Simplify each complex rational expression. (1/x-1/2)/(1/3-x/6)

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1. Rewrite the complex rational expression as a single fraction in the numerator and a single fraction in the denominator. The numerator is \( \frac{1}{x} - \frac{1}{2} \), and the denominator is \( \frac{1}{3} - \frac{x}{6} \).
2. Find a common denominator for the terms in the numerator. The least common denominator (LCD) of \( x \) and \( 2 \) is \( 2x \). Rewrite \( \frac{1}{x} \) as \( \frac{2}{2x} \) and \( \frac{1}{2} \) as \( \frac{x}{2x} \). Combine the terms: \( \frac{2}{2x} - \frac{x}{2x} = \frac{2 - x}{2x} \).
3. Find a common denominator for the terms in the denominator. The least common denominator (LCD) of \( 3 \) and \( 6 \) is \( 6 \). Rewrite \( \frac{1}{3} \) as \( \frac{2}{6} \) and \( \frac{x}{6} \) remains as is. Combine the terms: \( \frac{2}{6} - \frac{x}{6} = \frac{2 - x}{6} \).
4. Rewrite the complex fraction as a division problem: \( \frac{\frac{2 - x}{2x}}{\frac{2 - x}{6}} \). Division of fractions is equivalent to multiplying by the reciprocal, so rewrite this as \( \frac{2 - x}{2x} \times \frac{6}{2 - x} \).
5. Simplify the expression by canceling out the common factor \( 2 - x \) in the numerator and denominator (assuming \( 2 - x \neq 0 \)). Multiply the remaining terms: \( \frac{6}{2x} \). Simplify further if possible.

Key Concepts

Complex Rational Expressions

A complex rational expression is a fraction where the numerator, the denominator, or both contain rational expressions. To simplify these expressions, one must often find a common denominator for the fractions involved, which allows for easier manipulation and simplification of the overall expression.

Finding a Common Denominator

Finding a common denominator is essential when adding, subtracting, or simplifying fractions. The common denominator is a shared multiple of the denominators of the fractions involved, which allows for the fractions to be combined or simplified effectively. In the context of complex rational expressions, this step is crucial for simplifying the numerator and denominator.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms by dividing the numerator and denominator by their greatest common factor (GCF). This process is important in rational expressions to make them easier to work with and to clearly present the final result. In complex rational expressions, simplification can lead to a clearer understanding of the relationship between the variables involved.