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Solve the following for $x:$ $\left\{\begin{matrix} 3^x > 9 \\ 3^{x^2} \leq 27^{4x}. \end{matrix}\right.$

What is the range of $x$ that satisfies the inequality $\left ( \frac{1}{49} \right )^{x+4} + \left ( \frac{1}{7} \right )^{x+4} > 12 ?$

What is the range of $x$ that satisfies the inequality $\left(\frac{1}{5}\right)^{3+x} < 625 < \left( \frac{1}{25} \right)^x ?$

Given $\log 2=0.3010$ and $\log 3=0.4771,$ solve the inequality $\left(2 ^{7}\right)^x<\left(3 ^x\right)^{6}.$

Given $n \geq 2,$ which of the following numbers is the largest: $\sqrt[n]{6^{n-1}}, \sqrt[n]{6^{n+1}}, \sqrt[n+1]{6^n},6 ?$

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