In a book i'm writing, i want to introduce students to equations in slightly more general terms. Solving an equation with some unknown is just one example of finding an object by fact that it is a part of a certain equivalence class - specifically a class in the "equals"-relation. Same goes for inequalities too. In both scenarios we use various operations on both sides, make sure that none of these operations change the equivilance class of any member, and then isolate some desired expression. This is really intuitive and natural to work with for equality, but i think it could be a great exercise for the mind and maturity to try something very different. Does there exist any other nice similair problems and techniques for different relations? Maybe something using congruence or similarity of figures? We can obviously work with set equalities too, though that may be a little very abstract.
For information, my readers has not yet at this point in the book learned calculus. It's at just the point before. The target audience is math-interested high-schoolers.