solve and write the solution in interval notation: $\frac{1}{3} + \frac{6}{x^2}

Question

solve and write the solution in interval notation: $\frac{1}{3} + \frac{6}{x^2} < \frac{3}{x}$.

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Inequalities", "Interval Notation" ], "new_concepts": [], "current_concepts": [ "Rational Inequalities", "Interval Notation" ] } ## Find the critical values by setting the inequality to zero $$ \begin{aligned} &\frac{1}{3} + \frac{6}{x^2} - \frac{3}{x} < 0 \ &\frac{x^2 - 9x + 18}{3x^2} < 0 \ &\frac{(x-3)(x-6)}{3x^2} < 0 \end{aligned} $$ ## Determine the sign of the expression on each interval $$ \begin{aligned} &\text{Undefined at: } x = 0 \ &\text{Zeroes at: } x = 3, \quad x = 6 \ &\text{Intervals to test: } (-\infty, 0), \quad (0, 3), \quad (3, 6), \quad (6, \infty) \ &\text{Since } 3x^2 > 0 \text{ for all } x eq 0, \text{ the sign depends on } (x-3)(x-6): \ &\text{For } x \in (3, 6), \quad (x-3)(x-6) < 0 \end{aligned} $$ ## Write the solution in interval notation $$ x \in (3, 6) $$ Solve and write the solution in interval notation: $\frac{1}{3} + \frac{6}{x^2} < \frac{3}{x}$. Provide your answer below: $(3, 6)$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Rational Inequalities" ] }

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Question

Explanation:

Find the critical values by setting the inequality to zero

Determine the sign of the expression on each interval

Write the solution in interval notation

Answer: