Solve Inequality x+3^x<4 | Logical & Analytic Ways

[Karim Habashy]

Hi all,

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

Thanks

 

[Mentallic]

There is no analytic solution to finding the zeroes of the equation x+3^x-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?

 

[Karim Habashy]

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]

Sorry about the late response, I guess I missed this thread in my alerts.

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.

 

[HomogenousCow]

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

 

[Karim Habashy]

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

 

[HallsofIvy]

By stating that the function is monotonically increasing, you have used Calculus. And the fact that it is monotonically increasing leads to Homogeneous Cow's point, that there can only be one point where f(x)= 4.

 

[LAZYANGEL]

Trial and error is the way to go, unless you want to put the log product function into play.