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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.

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What is the solution set to \[\frac{1}{32}\leq\left(\frac{1}{2}\right)^{2x+1} \leq 4?\]

What is the solution set to \[0.5^{x+4}\geq16?\]

If \(0<a<b,\) the solution set to \[a<a^xb^{1-x}<b\] is \(\alpha<x<\beta.\) Find \(-2\alpha+3\beta.\)

What is the smallest integer \(x\) that satisfies \[\left(\frac{1}{2}\right)^x<16?\]

If the solution to \[\frac{1}{27}<\left(\frac{1}{9}\right)^x<2187\] is \(\alpha<x<\beta,\) what is \(\alpha+\beta?\)

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