If you mean (3x)ˣ⁺¹-(3x)ˣ⁻¹=8, then x=1 is a solution: 3²-1=8.
If you mean 3(xˣ⁺¹)-3(xˣ⁻¹)=8, then the solution can be found using the Intermediate Value Theorem.
Let f(x)=3(xˣ⁺¹)-3(xˣ⁻¹)-8, then:
f(0)=-8, f(1)=-8, f(2)=3(8)-3(2)-8=10.
So there must be a value of x between 1 and 2, where f(x)=0, because f(x) goes from negative to positive through zero.
f(1.5)=-3.4..., so f(x)=0 for x between 1.5 and 2. f(1.75)=1.4... Therefore x is between 1.5 and 1.75.
And so on. Continuing in this way we find x=1.6901 approx.