Proving Inequalities
Big Idea: Proving an inequality means showing, using logical steps and algebraic reasoning, that one expression is always greater than (or less than) another under given conditions. Unlike solving an inequality (finding values of $x$), proving an inequality means demonstrating it is always true.
1. What Does It Mean to "Prove" an Inequality?
When we solve $2x + 3 > 7$, we find $x > 2$. But when we prove an inequality like $a^2 + b^2 \geq 2ab$, we show it's true for all real numbers $a$ and $b$. No exceptions!
Key Principle
A perfect square is always non-negative: for any real number $x$,
$$x^2 \geq 0$$
This simple fact is the foundation of most inequality proofs at this level!