question

solve: $\frac{1}{x - 4} = \frac{x - 3}{6}$. if there are multiple solutions, list them separated by commas, e.g. $x = 1, 2$. if there is no solution, enter $\varnothing$.

provide your answer below:

$x = \square$

Question

question

solve: $\frac{1}{x - 4} = \frac{x - 3}{6}$. if there are multiple solutions, list them separated by commas, e.g. $x = 1, 2$. if there is no solution, enter $\varnothing$.

provide your answer below:

$x = \square$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [], "current_concepts": [ "Solving Rational Equations", "Solving Quadratic Equations" ] } ## Cross-multiply to eliminate fractions $$ \begin{aligned} \frac{1}{x-4} &= \frac{x-3}{6} \quad (x eq 4) \ 1 \cdot 6 &= (x-4)(x-3) \ 6 &= x^2 - 7x + 12 \end{aligned} $$ ## Solve the quadratic equation $$ \begin{aligned} x^2 - 7x + 6 &= 0 \ (x-1)(x-6) &= 0 \ x = 1 \quad &\text{or} \quad x = 6 \end{aligned} $$ ## Verify the solutions against the domain restriction $$ \begin{aligned} x & eq 4 \ x &= 1, 6 \end{aligned} $$ Solve: $\frac{1}{x-4} = \frac{x-3}{6}$. If there are multiple solutions, list them separated by commas, e.g. $x = 1, 2$. If there is no solution, enter $\varnothing$. Provide your answer below: $x =$ $1, 6$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Solving Rational Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Cross-multiply to eliminate fractions

Solve the quadratic equation

Verify the solutions against the domain restriction

Answer: