Proof of Inequalities

Proof of inequalities is a key part of the Proof topic in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-01). This subtopic proves inequalities from the definition \(a>b\) when \(a-b>0\), from the fact that squares are non-negative such as \((a-b)^2 \ge 0\), and using the triangle inequality \(|a+b| \le |a|+|b|\) and the arithmetic mean–geometric mean inequality \(\dfrac{a+b}{2} \ge \sqrt{ab}\).

Students and teachers will cover how to build new inequalities from previously known ones, prove inequalities involving geometry, and prove inequalities using graphical or calculus techniques including the squeeze theorem — a demanding skill rewarded in the HSC Extension 2 course.