$\infty = -1 $ paradox

When we talk about an “infinite sum”, we are really talking about a limit. In this case, we are talking about the limit of the “partial sums” of the series. The partial sums are:
$$\begin{align*}
s_1 &= 1;\\
s_2 &= 1+2;\\
s_3 &= 1+2+4;\\
&\vdots
\end{align*}$$
That is, $s_n$ is the sum of the first $n$ summands in the series. When we talk about the “value” of a series (an infinite sum), we are really talking about the limit of the $s_n$: that is, a specific real number $L$ that the $s_n$ are approaching as $n\to\infty$. Or we say that a series “equals $\infty$” if the values of $s_n$ grow without limit.

When you say $x = 1+2+4+\cdots$, what you are really saying is that the limit of $s_n$. In this case, the limit of the $s_n$ does not exist, because
$$\lim_{n\to\infty}s_n = \infty.$$
The values of $s_n$ get arbitrarily large as $n\to\infty$.

It is certanly true as well that the sum $2+4+8+\cdots$ is also $\infty$, since $2\times\infty = \infty$ (in the extended reals). And if you subtract one, then you still get $\infty$ because $\infty -1 = \infty$ (in the extended reals).

So you can write $2x = x-1$.

What you cannot do, however, is “subtract $x$ from both sides”; because that would be writing
$$2\times\infty – \infty = \infty -1 -\infty$$
and the problem is that even in the extended reals, $\infty-\infty$ is undetermined. It does not equal anything, and certainly not zero. In short, you cannot just cancel infinities.