Solve for $x$: $x+3^x<4$ I was doing some questions for Calculus 3rd edition by Michael Spivak, and in one the questions for the prologues was to solve for $x$:
$$x+3^x<4$$
I wasn't sure how to simplify this questions, should I assume we know more than the postulates given previously (addition, multiplication, inequalities)?
 A: (This is largely a repost of comments I made here.)
Echoing Joe's sentiments, IMO this part of the exercise (1-4.(xii), 3rd ed.) should not have been included in the chapter. The problem is at odds with the rigour Spivak's trying to encourage elsewhere. This also goes for the two preceding parts (x) and (xi), each of which similarly involve expressions beyond the scope of what Spivak carefully lays out in the chapter ($\sqrt[3] 2$ and $2^x$, respectively.)
Arguments based on derivatives, strictly increasing functions, continuity, etc. cannot be expected of the reader, and yet without these tools, the reader really has no way of definitively answering the problem at this point in the text.
Per $x+3^x<4$ specifically, it's fair to assume readers will have previous familiarity with these things and be able to guess the answer. Even so, they have no way of proving that their answer is correct.
All that said, don't let this put you off @Yanbo. Keep going! The rewards in the book far outweigh the few hiccups. I suggest keeping a list of suspected errors and mistakes you find during your reading, and continue to check in here at MSE, which functions as sort of a defacto errata list for Spivak.
(Note, this comes up again here as well. The search function on MSE leaves much to be desired. One trick that seems to help: when typing your question, if you enter some key formula into the question title field, this should bring up a list of questions containing similar text, some of which might match your own question. This doesn't appear to work very well for this particular question though, for some reason.)
A: In this context, you are indeed allowed to assume more than the "basic properties of numbers" that have already been introduced. Here, it is valid to assume that $3^x$ has been defined for all rational values of $x$, where $3^{a/b}$ can be defined as $(\sqrt[b]{3})^a$. If $x$ is negative, then $3^x=1/(3^{-x})$. At this point, it is not clear what $3^{x}$ should mean for irrational $x$, and so you can limit yourself to rational exponents. Irrational exponents are covered in Chapter $18$.
There are other instances of this in the book: the trigonometric functions are often used in examples and exercises, even though they are only given a rigorous treatment in Chapter $15$; likewise, there are exercises which assume the existence of square roots of nonnegative real numbers, even though this is only justified in Chapters $7$ and $8$. Ideally, you should be able to do the exercises using only the material that has been introduced thus far. However, this is sometimes not possible, especially in the earlier chapters. If you are unsure about what you are allowed to assume, you might want to consult the solutions at the back of the book or the solution manual.
