Disequazione irrazionale $\sqrt{2x+1}<x-1$

Text

Solve the irrational inequality $\sqrt{2x+1}<x-1.$

Solution

An inequality of the form $\sqrt{A}<B$ is equivalent to the system $A\ge 0,\ B>0,\ A<B^2$.

• $2x+1\ge 0\implies x\ge -\tfrac12$;
• $x-1>0\implies x>1$;
• $2x+1<(x-1)^2=x^2-2x+1\implies 0<x^2-4x\implies x(x-4)>0\implies x<0\ \vee\ x>4$. Intersecting the three conditions ($x>1$ removes the branch $x<0$): $\boxed{x>4.}$