# Inequality x+3^x<4

Hi all,

I was trying to solve the Inequality x+3x<4 and i found the solution to be x<1, using trial and error.
Is there another Logical way or analytic one.

Thanks

Mentallic
Homework Helper
There is no analytic solution to finding the zeroes of the equation x+3x-4, hence there is no solution other than by numerical or observational means to solve yours (unless you're willing to use the Lambert W function). What you can do however is to prove that x<1 is the only possibility. What if there are other values of x > 1 that work? Can you prove there aren't?

Ya, strengthening my answer by contradiction is a good idea (i.e what happens if x>1).
I will also have a look at Lambert W Function.

Thanks

Mentallic
Homework Helper

Just to be clear, proving that the equation cannot hold for x>1 is NOT a proof by contradiction. You can't automatically assume that if x>1 does not satisfy the inequality, then x<1 must satisfy it. It doesn't work that way.

I'd suggest calculus for showing that there are no values x>1 that satisfy the equation.

You could simply observe that x+3^x=4 at x=1, and argue that since x+3^x has a positive gradient everywhere, all x which satisfy x<1 must satisfy x+3^x<4

How do i use calculus ?, the function is monotonically increasing.

HallsofIvy