I'm a little rusty, cant solve x+e^x=b

[allamid06]

Homework Statement

Hi, I'm new here. I'm really rusty, I resume my career this year, and I'm reading 'the spivak book', (for Calculus 1).
Making some exercises, I get curious about how to solve this: x+e^x=4
I would love if someone could give me any trick

Homework Equations

The Attempt at a Solution

 

[jedishrfu]

Welcome to PF!

I don't think there is an analytical way to solve for x in this example. I did graph it and found that x=1.0737... approximately.

You could find this by trying different values of x like 0,1,2... And then narrowing in the values. You could also uses a Newtons approximation formula or a graphing calculator to get the result.

Try doing it on a graphics calculator or using some graphing software like freemat on a PC or Mac.

 

[HallsofIvy]

For an "analytical solution", you can first write the equation as e^x= 4- x, then divide both sides by e^x to get (4- x)e^{-x}= 1. Now, let y= 4- x so that -x= y- 4. In terms of y, the equation is ye^{y- 4}= ye^y(e^{-4})= 1. Multiply both sides of the equation by e^4 to get ye^y= e^4.

Finally, apply the "Lambert W function", which is defined as "the inverse function to f(x)= xe^x", to both sides of the equation getting y= W(e^4). Since x= 4- y, the solution to the original equation is x= 4- W(e^4).

 

[allamid06]

For an "analytical solution", you can first write the equation as e^x= 4- x, then divide both sides by e^x to get (4- x)e^{-x}= 1. Now, let y= 4- x so that -x= y- 4. In terms of y, the equation is ye^{y- 4}= ye^y(e^{-4})= 1. Multiply both sides of the equation by e^4 to get ye^y= e^4.

Finally, apply the "Lambert W function", which is defined as "the inverse function to f(x)= xe^x', to both sides of the equation getting y= W(e^4). Since x= 4- y, the solution to the original equation is x= 4- W(e^4).

Thanks Hallsoflvy, and thanks jedishrfu. First of all, I will get the Hallsoflvy answer, because, I'm pretending to use anything but the first cap, of the Spivak.
By the other part, Hallsoflvy, I did that, but then I realize, that Lambert W function was too over the first cap of the book. Anyway I really like the way you explain me that method.
To be honest, the original problem was x+3^{x}<4. I use desmos.com a lot, to graph these things. I could solve this by seeing that 3^{x}=3 only if x=1, and since 3^{x} rises, (by "common sense"), 3^{x}<3 when x>1 and at the same time I can add these and get x+3^{x}<1+3. I could apply the same to show x+3^{x}>4 when 1>x.
But when I was doing this, I just get curious about how would I find the root, in this case. And then I went (don't know why) to x+e^{x}=b thinking it would be easier.
PD:Thanks for your time. Since I study software engineer, I will be more interested in Calculus, Discrete Maths,
Probability and Statistics etc. But also I will have Pysics I. What I'm saying is that I hope to be useful in this community, and I hope, there where place to my questions.
Also I'm from Uruguay, my apologies for my english. -I can understand you but I'm not really good writing-