It is worthy of remark that there is a method of proving algebraic inequalities which is very generally applicable, and which furnishes proofs of great directness and completeness.

The method consists in expressing the difference A – B in a manifestly positive form, when we have to prove A > B.

The first inequality occurring in school Algebra is usually proved in this way. We have to show that if a, b, and x are positive quantities is nearer to 1 than . We have

The two cases a > b and b > a are then considered separately, and the conclusion drawn.