\\frac{\frac{x^2-4}{x-2} + \frac{x^2-9}{x-3}}{\frac{x^2-1}{x-1} - \frac{x^2-16}{x-4}} \quad (x \
eq 2, 3, 1, 4)\

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Difference of Squares", "Rational Expressions", "Simplifying Fractions" ], "new_concepts": [], "current_concepts": [ "Difference of Squares", "Rational Expressions", "Simplifying Fractions" ] } ## Simplify the numerator and denominator terms using difference of squares $$ \begin{aligned} \frac{x^2-4}{x-2} &= \frac{(x-2)(x+2)}{x-2} = x+2 \quad (x eq 2) \ \frac{x^2-9}{x-3} &= \frac{(x-3)(x+3)}{x-3} = x+3 \quad (x eq 3) \ \frac{x^2-1}{x-1} &= \frac{(x-1)(x+1)}{x-1} = x+1 \quad (x eq 1) \ \frac{x^2-16}{x-4} &= \frac{(x-4)(x+4)}{x-4} = x+4 \quad (x eq 4) \end{aligned} $$ ## Substitute simplified terms into the complex fraction $$ \begin{aligned} \text{Numerator} &= (x+2) + (x+3) = 2x + 5 \ \text{Denominator} &= (x+1) - (x+4) = x + 1 - x - 4 = -3 \end{aligned} $$ ## Write the final simplified expression $$ \frac{2x+5}{-3} = -\frac{2x+5}{3} $$ $$ -\frac{2x+5}{3} $$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Rational Expressions" ] }

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Question

Explanation:

Simplify the numerator and denominator terms using difference of squares

Substitute simplified terms into the complex fraction

Write the final simplified expression

Answer: