Exponential inequality with base below one

Text

Solve the inequality $\left(\dfrac{1}{2}\right)^{x+1}>8$.

Solution

Write $8$ as a power of the base $\tfrac12$: $8=2^{3}=\left(\tfrac12\right)^{-3}$. The inequality becomes $\left(\tfrac12\right)^{x+1}>\left(\tfrac12\right)^{-3}.$ The base $\tfrac12$ is less than $1$, so the exponential is decreasing: passing to exponents reverses the direction, $x+1<-3\;\Rightarrow\; x<-4.$ $\boxed{\,x<-4\,}$