Math Problem Statement

Simplify the expression: \( \frac{3 + 6 \left( \frac{3}{x-6} \right)}{\frac{3}{x-6}} \)

Solution

The given expression is:

3+6(3x6)3x6\frac{3 + 6 \left( \frac{3}{x-6} \right)}{\frac{3}{x-6}}

Step-by-Step Solution:

  1. Simplify the numerator: 3+6(3x6)=3+18x63 + 6 \left( \frac{3}{x-6} \right) = 3 + \frac{18}{x-6} This can be rewritten as a single fraction: 3+18x6=3(x6)x6+18x6=3x18+18x6=3xx6.3 + \frac{18}{x-6} = \frac{3(x-6)}{x-6} + \frac{18}{x-6} = \frac{3x - 18 + 18}{x-6} = \frac{3x}{x-6}.

    So the numerator becomes 3xx6\frac{3x}{x-6}.

  2. Rewrite the full expression: The original expression now looks like this: 3xx63x6.\frac{\frac{3x}{x-6}}{\frac{3}{x-6}}.

  3. Simplify the division: Division of fractions is performed by multiplying the numerator by the reciprocal of the denominator: 3xx63x6=3xx6x63.\frac{\frac{3x}{x-6}}{\frac{3}{x-6}} = \frac{3x}{x-6} \cdot \frac{x-6}{3}.

    Cancel the (x6)(x-6) terms (assuming x6x \neq 6) and simplify: 3x3=x.\frac{3x}{3} = x.

Final Answer:

xx

Let me know if you would like a more detailed breakdown of any step!


Follow-Up Questions:

  1. Why is it important to check the domain of xx (e.g., x6x \neq 6) in this problem?
  2. How do we handle simplifying fractions within fractions effectively?
  3. What other algebraic techniques can be used to simplify complex rational expressions like this one?
  4. How does dividing fractions differ from multiplying fractions?
  5. How can verifying the solution improve confidence in algebraic manipulations?

Tip:

Always check for restrictions in the domain (e.g., x60x-6 \neq 0) when working with rational expressions to avoid undefined results.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplifying Rational Expressions
Fractions

Formulas

Simplifying complex fractions by multiplying by the reciprocal
Combining like terms in rational expressions

Theorems

Fraction Division Rule

Suitable Grade Level

Grades 9-11