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 \[\left(\frac{1}{2}\right)^x<16?\]
If the solution to \[\frac{1}{27}<\left(\frac{1}{9}\right)^x<2187\] is \(\alpha<x<\beta,\) what is \(\alpha+\beta?\)