Upside-down equation: algebra puzzle

Question

This quirky equation was presented to me by a fellow teacher, along with the instruction "solve it both ways up".

The equation is simple enough to solve, with one integer root. What's more interesting is that the page can be rotated $180$° and a new equation can be read.

This second equation has two roots. I found it fairly satisfying that both of its roots are integers, but I was hoping that it would share a root with the first equation.

I set about looking for an equation that makes sense when read upside-down, where the two equations share a root.

My students immediately came up with trivial examples like "$x=1+1$" and "$1=\frac{1}{x}$". I haven't found anything more interesting yet. I haven't even figured out an approach better than trial and error, with different permutations of the digits $1$, $6$, $8$ and $9$.

Can anybody help me to find such an equation, ideally one that is similarly "difficult"/"interesting" as the equation presented to me?

Edit: credit is due to Rob Eastaway who (I have learned since posting this question) originally posted the first image on Twitter.

Answer 1

The answers by @caverac and @Bram28 very much helped me to come to some answers of my own.

My method has been to guess at the form of an equation, such as $ \frac{x+a}{b} + c = \frac{x+1}{d}$, which can be rotated to give $ \frac{d'}{1+x}=c'+\frac{b'}{a'+x} $, where $a$, $b$, $c$ and $d$ are constants that can be read as $a'$, $b'$, $c'$ and $d'$ when rotated.

I then used a fairly simple Excel spreadsheet to search for equations of this form that share solutions with their rotated forms, with the constants being selected from the set of all one or two digit numbers that can be read upside down:

$$S=\{1,8,6,9,11,18,81,16,61,19,91,88,86,68,89,98,66,99,69,96\}$$

(I have not included $5$ because my own written $5$s don't look like $5$s when upside down.)

The above form delivered no results. Neither did the forms $\frac{x}{a} =b- \frac{1-x}{c}$ or $\frac{a-x}{b} +c= \frac{1+x}{d}$.

But the form $ \frac{x+a}{b} =c- \frac{x-1}{d}$ delivered the equation:

$$ \frac{x+61}{6} =1- \frac{x-1}{8}$$

which shares an integer root with its rotated form, and:

$$ \frac{x+61}{8} =6- \frac{x-1}{16}$$

which shares a rational root with its rotated form.

I guess that more examples could be found by trying other equation forms; by including three digit numbers in $S$; or by allowing the constants to be sums of numbers from $S$, for example $a=14=8+6$ with $a'=17=9+8$.

Answer 2

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

Answer 3

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

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

Answer 4

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: