If you replace the equal sign of an equation with an inequality sign, is there ever a time when the same value will be a solution to both the equation and the inequality?

Answer 1

If inequality sign is referring to simply < or >, then the answer has to be no. This is even an axiom, i.e. one of the most fundamental concepts on which the whole building of math is built on. This implies our unability to prove the spoken-of axiom - it is a definition.

Answer 2

The axiom is called axiom of trichotomy.