Drop the zeros

$11111_{x}=1x^{0}+1x^{1}+1x^{2}+1x^{3}+1x^{4}\Rightarrow$
$11111_{x}=x^{4}+x^{3}+x^{2}+x+1$

We want:
$101010101=11111_{x}\Rightarrow$
$101010101=x^{4}+x^{3}+x^{2}+x+1\Rightarrow$
$x^{4}+x^{3}+x^{2}+x-101010100=0$

This equation has two roots : $100$ and $-100.50003...$ (According to Symbolab)
Since we want $x$ to be positive, the solution is:
$\boxed{x=100}$

Note : I understand that you wanted us to use your reasoning but i thought i'd share my solution.

$101010101_{10} = 11111_x$ Hence the deduction:

$x^4+x^3+x^2+x^1=101010100 \text{[See Maximos's solution]}$ $101010100=10^8+10^6+10^4+10^2=(10^2)^4+(10^2)^3+(10^2)^2+(10^2)^1$ So it is valid and consistent to conclude the value of $x$ to be $\boxed{10^2=100}$