Problem

Source: Kosovo National Olympiad 2025, Grade 9, Problem 4

Tags: absolute value, algebra, inequalities



Show that for any real numbers $a$ and $b$ different from $0$, the inequality $$\bigg \lvert \frac{a}{b} + \frac{b}{a}+ab \bigg \lvert \geq \lvert a+b+1 \rvert$$holds. When is equality achieved?