Use the graph of the combined function y=3^x - x^3 - 1 to determine an approximate solution to the inequality 3^x < x^3 + 1.
3^x < x^3 + 1
means
3^x - x^3 - 1 < 0
so, take a look at the graph and find where this is true.
https://www.wolframalpha.com/input/?i=3%5Ex+-+x%5E3+-+1+for+-1.5+%3C%3D+x+%3C%3D+3.5
Remove the domain specification to see more clearly where the graph crosses the x-axis (note the roots listed).