Is there a closed-form solution (even approximated) to this inequality?

Question

I have the following function:

$f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) \right]$

With $\theta > 0$ (with $\theta \ne \frac{1}{2},1$), $N > 1$ and $0 \le x < N$.

I am facing two problems:

1. I am trying to solve $f(x, \theta) = 0$ and $f(x, \theta) \ge 0$ in terms of $x$. Is there a closed form solution (even approximated) for this, assuming, if needed, that $N$ is large and $x \ll N$?

2. I am also trying to solve $f(0, \theta) \ge 0$ in terms of $\theta$.

   $f(0, \theta) = (1-\theta)\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta}) - (N^{1-\theta} - 1) \right]$

   Is there a closed-form solution (even approximated) to solve this, assuming, if needed, that $N$ is very large and $x \ll N$?

Thank you very much in advance.