Always True, Never True

Sometimes the x disappears. x + 1 < x + 2: subtract x from both sides and 1 < 2 is left, with no x in sight. It is true, and it was true whatever x was: every number is a solution. The solution set is all real numbers.

x > x + 1: subtract x, and 0 > 1 is left, which is false for every x. No number is a solution.

So when the x-terms cancel, look at what remains: a true statement means every number works, a false one means none does. Neither is a mistake. A number is always one less than itself plus one, and never more than itself plus one.