In Exercises 69–82, simplify each complex rational expression. (x/3−1)/(x−3)

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Step 1: Identify the complex rational expression. The given expression is \( \frac{\frac{x}{3} - 1}{x - 3} \). A complex rational expression is a fraction where the numerator or denominator (or both) contain fractions.

Step 2: Simplify the numerator \( \frac{x}{3} - 1 \). To combine \( \frac{x}{3} \) and \( -1 \), rewrite \( -1 \) with a denominator of 3: \( -1 = \frac{-3}{3} \). Now, the numerator becomes \( \frac{x}{3} - \frac{3}{3} = \frac{x - 3}{3} \).

Step 3: Rewrite the entire complex rational expression using the simplified numerator. The expression now becomes \( \frac{\frac{x - 3}{3}}{x - 3} \).

Step 4: Simplify the complex fraction by dividing the numerator by the denominator. Recall that dividing fractions involves multiplying by the reciprocal. Rewrite \( \frac{\frac{x - 3}{3}}{x - 3} \) as \( \frac{x - 3}{3} \times \frac{1}{x - 3} \).

Step 5: Cancel out \( x - 3 \) in the numerator and denominator (assuming \( x \neq 3 \) to avoid division by zero). The simplified expression is \( \frac{1}{3} \).

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Key Concepts

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

Complex Rational Expressions

A complex rational expression is a fraction where the numerator, the denominator, or both contain rational expressions. Simplifying these expressions often involves finding a common denominator and reducing the fraction to its simplest form. Understanding how to manipulate these expressions is crucial for solving problems involving them.

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 essential in algebra to make expressions easier to work with and to clearly identify relationships between variables. Mastery of this concept is necessary for effectively simplifying complex rational expressions.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions, allowing for the addition, subtraction, or comparison of those fractions. In the context of complex rational expressions, finding a common denominator is often the first step in simplification, enabling the combination of terms and the reduction of the overall expression.