How to solve the following mixed exponential inequality: $3^x + x < 4$? (from Spivak's Calculus)

Question

I have encountered this inequality in Spivak's Calculus (first chapter exercises), which I'm not sure how to solve.

$$ 3^x + x < 4 $$

I might be wrong but my gut feeling says the inequality holds for all $x$ between $(-\infty, 1)$ but I cannot prove it.

As I read, there seems to be no standard scheme for solving this type of inequalities/equations. How do You then usually proceed when dealing with one like the above? Thanks.

Answer 1

Because $x$ and $3^x$ are increasing, so is $x+3^x$.

Answer 2

defining $$f(x)=3^x+x-4$$ then we get $$f'(x)=3^x\ln(3)+1>0$$ can you finish?