Upside-down equation: algebra puzzle

There's of course

$$ \frac{1 + x}{1} = \frac{1}{x} ~~~\mbox{and}~~~ \frac{x}{1} = \frac{1}{x + 1} $$

These two don't have any integer roots, but they have the golden ratio as one

If I may be allowed to use $2$ and $5$ as well (their digital representations can be flipped ... and result in themselves):

$$\frac{82-x}{59+61-x+51}=\frac{x-28}{65+x-19+15}$$

Both original and flipped version have $x=55$ as a solution

Here is a digitized version:

and its flipped version:

I think $0=(x-1)(x-1)$ does it.

Maybe $0=(x-6)(x-1)$ shares only one root.