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). Students show that \(a>b\) by proving \(a-b>0\), use the fact that a real square such as \((a-b)^2\) is never negative, and work with \(|a+b| \le |a|+|b|\) (the triangle inequality) and the AM–GM inequality \(\dfrac{a+b}{2} \ge \sqrt{ab}\) for \(a,b \ge 0\).

Students and teachers will cover chaining known inequalities to reach new ones, inequalities that arise in geometry (such as the side lengths of a triangle), and arguments that use graphs, derivatives or the squeeze theorem — demanding reasoning that rewards careful setting out in the HSC Extension 2 course.