Spivak's Calculus - Problem 1.4(xii) [exponential inequality]

[middleCmusic]

Homework Statement

The task is to find all solutions of the following inequality:

$x+3^x <4$

But I was trying to find a solution for this problem in general:

$x+a^x < b$

Homework Equations

n/a

The Attempt at a Solution

$a^x < b-x$

$\text{log}_a(a^x) < \text{log}_a (b-x)$

$x < \text{log}_a(b-x)$

I can't see how to isolate $x$...

[Context: I'm going through Spivak for self-study to patch up holes in my understanding (and for fun). I'm a 3rd-year undergrad so this should be easy for me, but I can't figure this one out. ]

 

[Dick]

You don't want to go to the general case and try to isolate x. You can't do it. You do want to realize that x+3^x is an increasing function. Where does it equal 4? Solve that by guessing.

 

[STEMucator]

You don't want to go to the general case and try to isolate x. You can't do it. You do want to realize that x+3^x is an increasing function. Where does it equal 4? Solve that by guessing.

Yes, this is the obvious route here.

I believe he was asking about the problem in general though. As in what if you we're trying to solve $x + a^x < b$. I looked at it for a moment and realized there is indeed no way to solve it algebraically unless you're able to find where the graphs intersect. This requires there to be some numbers involved sadly otherwise we can't really do anything at all as you have to guess at the intersection point.