This document discusses three methods for comparing functions using inequalities: 1) Compare functions visually using different colors or styles, 2) Analyze the difference function by graphing it to see where it is above the x-axis, and 3) Analyze the quotient function by graphing it to identify where one quotient is greater than the other, corresponding to where one function is greater than the other. The document also covers using the quadratic formula with inequalities and applying these methods to basic functions and word problems.