Am I able to solve $xe^x<1$?

Question

Can I solve manually the inequality $xe^x<1$? Or any other exponential times polynomial inequality.

Answer 1

The solution to the equation $xe^x=1$ is famously known as the Omega constant, and for $x<\Omega$, we have

$$xe^x<1$$

For linear functions and exponential functions, we have the Lambert W function,

$$p^x=ax+b\implies x=\frac{W\left(-\frac{\ln p}ap^{-b/a}\right)}{-\ln p}-\frac ba$$

However, with the exception of special cases like $(x+1)^2e^x=4$ which may be reduced to a linear function times an exponential function via square rooting and such, I know not of a general solution in terms of special functions when it does not reduce to a linear function times the exponential function.