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# Exponential Inequalities

When will a bacteria colony reach a certain size? When will one investment outperform another? Many real-world models use exponential functions, and you'll need these tools to compare them.

Which of these numbers is larger?

\[ \large A = \frac{2^8}{5^8} \hspace{0.3cm} \text{ , } \hspace{0.3cm} B = \frac{2^{7}}{5^{7}} \]

What values of \(x\) satisfy \[\large 3^{x+ 2} < 3^{10}?\]

What values of \(x\) satisfy \[\large \frac{3^{x+ 5}}{3^{11}} > 1?\]

What values of \(x\) satisfy \[\large \left(\frac{1}{3}\right)^x < 3^{x+5}?\]

True or False?

There is a real number \(x\) such that \[\large x < x^3 < x^4 < x^2.\]

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