Solve equation with exponents of x?

I might be wrong, but I don't think there's any closed solution to this.

You can find a numerical solution -- e.g., by setting up an Excel spreadsheet and checking the values as a function of x. The solution appears to be

x <= K

where K is approximately equal to 0.1536,

but this is an approximate solution, not an exact one.

If you were looking for a closed solution, here's what you might do:

1) Rewrite the inequality as follows:

2^(-5x) - 2^(12x) +3 >= 0

2) Substitute y=2^x:

y^(-5) - y^12 + 3 >= 0

Because y=2^x, we are looking for solutions with positive values of y. We can transform the inequality into a standard polynomial by multiplying it by y^5:

1 + 3y^5 - y^17 >= 0

This is a lot simpler than what we started with, but not simple enough. We'd like to find a positive value of y such that the left side exactly equals 0, but that's a 17th-order polynomial with no general way of solving it. If the exponents 5 and 17 had common factors, we could use substitution to simplify the inequality further, but they don't have common factors.

Hence (and again, I could be wrong), I don't think there's a solution in closed form. You have to resort to numerical methods. For instance, you could use the Newton-Raphson method to get increasingly close approximations to K. I used a cheap-and-dirty method (an Excel spreadsheet) instead.

Perhaps there's a clever way of factoring the left-hand side, but I don't see a solution offhand.

Incidentally, here's a high-accuracy solution from my HP calculator:

y_root = 1.11232186117

which corresponds to

x_root = 0.15357430634 (which is a higher-accuracy version of the value of K given above).

Thus, the solution to your inequality is

x <= 0.153574306...