Numerical Inequalities

An inequality is a statement that compares two quantities and shows that one is less than, greater than, or otherwise not equal to the other, using the symbols $>$, $<$, $\ge$, or $\le$. When any two real numbers are related by one of these symbols, the result is called a numerical inequality — for example, $9 < 11$ or $18 > 17$.

One important application connects the arithmetic mean (AM) and geometric mean (GM) of two positive numbers $a$ and $b$: their AM is $A = \dfrac{a+b}{2}$ and their GM is $G = \sqrt{ab}$. The AM–GM relationship states that the arithmetic mean of any two positive numbers is never smaller than their geometric mean:

Equality holds only when $a = b$, since $A - G$ can be shown to equal $\dfrac{(\sqrt{a}-\sqrt{b})^2}{2}$, which is always non-negative.

Working with inequalities also requires care about how algebraic operations affect the inequality sign. Adding or subtracting the same quantity from both sides never changes the direction of an inequality, but multiplying or dividing both sides by a quantity does matter: doing so by a positive number preserves the direction, while doing so by a negative number reverses it. These behaviours give a small set of standing properties for any real numbers $a$, $b$, $c$: