2. solve \$3^{4x} > 9^{x-3}\$ for \$x\$. \$x > -\\frac{1}{3}\$ \$x = \\frac{1}{3}\$ \$x > -1\$ \$x > 3\$

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponent Rules", "Solving Linear Inequalities" ], "new_concepts": [ "Exponential Inequalities" ], "current_concepts": [ "Exponent Rules", "Solving Linear Inequalities", "Exponential Inequalities" ] } ## Express both sides with a common base \[ \begin{aligned} &3^{4x} > (3^2)^{x-3} \\ &3^{4x} > 3^{2(x-3)} \end{aligned} \] ## Set up the inequality for the exponents \[ \begin{aligned} &4x > 2(x-3) \\ &4x > 2x - 6 \end{aligned} \] ## Solve for x \[ \begin{aligned} &2x > -6 \\ &x > -3 \end{aligned} \] (A) $x > -\frac{1}{3}$ (B) $x = \frac{1}{3}$ (C) $x > -3$ (D) $x > 3$ { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Inequalities" ] }

2. solve \$3^{4x} > 9^{x-3}\$ for \$x\$. \$x > -\\frac{1}{3}\$ \\…

QUESTION IMAGE

$2. solve \$3^{4x} > 9^{x-3}\$ for \$x\$. \$x > -\\frac{1}{3}\$ \\…$

Question

solve \$3^{4x} > 9^{x-3}\$ for \$x\$.

\$x > -\frac{1}{3}\$

\$x = \frac{1}{3}\$

\$x > -1\$

\$x > 3\$

Explanation:

Response

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"step_cot",
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"Exponent Rules",
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</pre_analysis>

Express both sides with a common base

\[

\begin{aligned} &3^{4x} > (3^2)^{x-3} \\ &3^{4x} > 3^{2(x-3)} \end{aligned}

\]

Set up the inequality for the exponents

\[

\begin{aligned} &4x > 2(x-3) \\ &4x > 2x - 6 \end{aligned}

\]

Solve for x

\[

\begin{aligned} &2x > -6 \\ &x > -3 \end{aligned}

\]
</reasoning>

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"question_type": "Multiple Choice",
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"Mathematics",
"Algebra",
"Exponential Inequalities"
]
}
</post_analysis>

Answer: