Find the sum of all the solutions to $2^{|x|} + 3|x| = 18.$
If $x$ is a solution, then $-x$ is a also a solution.  Thus, we can pair all the solutions, and their sum is $\boxed{0}.$

Let $f(x) = 2^{|x|} + 3|x|.$  Since $f(0) = 0$ and $f(4) = 28,$ the equation $f(x) = 18$ has at least one solution in the interval $0 \le x \le 4.$  This ensures that the sum that the problem asks for is not an "empty" sum.