Disequazione $(x-1)^2(x+2)<0$

Text

Solve the inequality $(x-1)^2(x+2)<0.$

Solution

The factor $(x-1)^2$ is always $\ge 0$ and vanishes at $x=1$. For the product to be strictly negative we need $(x-1)^2>0$ (i.e. $x\ne 1$) and at the same time $(x+2)<0$, i.e. $x<-2$. The condition $x\ne 1$ is automatically satisfied by $x<-2$. $\boxed{x<-2.}$