Always True, Never True

Sometimes the x-terms cancel out. In $x+1 < x+2$, subtract $x$ from both sides. What is left is $1 < 2$, with no $x$ in it. The statement $1 < 2$ is true, and it stays true whatever value $x$ has. So every number is a solution. The solution set is all real numbers.

In $x > x+1$, subtract $x$ from both sides. What is left is $0 > 1$, which is false whatever value $x$ has. So no number is a solution.

When the x-terms cancel, look at the statement that is left. If it is true, every number is a solution. If it is false, no number is a solution. Neither result means you made a mistake. Any number is always less than itself plus 1, and never greater than itself plus 1.