Can you write an equation by setting an expression equal to itself. would an equation like this be true

Of course! An expression is always equal to itself.

But if there's, say, an 'x' in the expression, then such an equation would be true

for x=every possible number. You might say it has an infinite number of solutions ...

which certainly has to be true, because no matter what you do to an expression,

it's still always equal to itself!

Yes it would be as long as it truly equals eachother