Overview

Inequalities are about bounding expressions. Learn to spot when AM-GM or Cauchy applies, and how to rewrite expressions to match known forms.

Key Ideas

• AM-GM: for nonnegative $a,b$, $\frac{a+b}{2}\ge\sqrt{ab}$.
• Cauchy-Schwarz: $(a^2+b^2)(x^2+y^2)\ge (ax+by)^2$.
• Equality cases often guide the right substitution.

Worked Example

For positive $a,b$, show $a+b \ge 2\sqrt{ab}$.

Apply AM-GM directly. Equality holds when $a=b$.

Practice Problems