Algebra
# Exponential Inequalities

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 $\left(\frac{1}{2}\right)^x<16?$