# How to solve the following mixed exponential inequality? (from Spivak's Calculus)

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.

Because $x$ and $3^x$ are increasing, so is $x+3^x$.
• You should know it from the properties of elementary functions ($a^x$ is strictly increasing for $a>1$). Moreover, it looks as an exercise before introducing derivatives. Am I right? And you should not expect a general method. Changing the constatnt from 4 to 5 makes the exercise unsolvable in a simple way. – Przemysław Scherwentke Mar 26 '18 at 9:14
• In the case 4 it is easy to guess that $x=1$ is the solution of equality. If you have 5, you can only estimate the solution numerically. – Przemysław Scherwentke Mar 26 '18 at 9:33
defining $$f(x)=3^x+x-4$$ then we get $$f'(x)=3^x\ln(3)+1>0$$ can you finish?