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Exponential Inequalities

Exponential Inequalities - Multiple Terms

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

\[ {3} < x \leq {13} \] \[ {4} \leq x < {14} \] \[ {5} \leq x < {15} \] \[ {2} < x \leq {12} \]

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by Brilliant Staff

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

\[ x < {4}+\log_{7} {3} \] \[ x < {4}+\log_{7} {11} \] \[ x < {-4}-\log_{7} {11} \] \[ x < {-4}-\log_{7} {3} \]

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by Brilliant Staff

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

\[ {-8} < x < {-3} \] \[ {-6} < x < {-1} \] \[ {-7} < x < {-2} \] \[ {-9} < x < {0} \]

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by Brilliant Staff

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

\[x>0\] \[x>1\] \[x>\log 2\] \[x>\log 3\]

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by Brilliant Staff

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

\[ \sqrt[n]{{6}^{n-1}} \] \[ {6} \] \[ \sqrt[n]{{6}^{n+1}} \] \[ \sqrt[n+1]{{6}^{n}} \]

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by Brilliant Staff

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