If any two integers are unequal \(x\neq y\), then the inequality is preserved, if and only if we add an arbitrary integer \(z\) to both sides of the inequality, formally: \[x \neq y\Longleftrightarrow \begin{cases} z + x\neq z + y,&\text{or}\\ x + z\neq y + z. \end{cases}\]
Proofs: 1
