Q.Insert the appropriate sign of inequality: $\sqrt{3}(\sqrt{50} - \sqrt{32}) \underline{\quad\quad} 3\sqrt{54} + 2\sqrt{24}$

The AM–GM Inequality: Why Averages Are Not All Equal

You already know what an average is — add up a few numbers and divide by how many there are. That is the arithmetic mean (AM). For two numbers $a$ and $b$, it is

Now imagine you want a different kind of "middle" — one that works for ratios, growth rates, or areas. If you multiply the numbers together and take the $n$th root, you get the geometric mean (GM). For two numbers $a$ and $b$,

The geometric mean answers a different question: "If I had a rectangle of sides $a$ and $b$, what side length would give me the same area in a square?" That square's side is $\sqrt{ab}$.

The Core Idea

Here is the surprising fact: the geometric mean is never larger than the arithmetic mean. They are equal only when all the numbers are identical. For any two non‑negative numbers $a$ and $b$,

with equality if and only if $a = b$.

This is the AM–GM inequality. It is one of the most used inequalities in mathematics — not because it is complicated, but because it is simple and powerful.

Why Should You Believe It?

Take any two non‑negative numbers, say $4$ and $9$.

• Arithmetic mean: $(4+9)/2 = 6.5$
• Geometric mean: $\sqrt{4 \times 9} = \sqrt{36} = 6$

Indeed $6.5 > 6$. Try $1$ and $100$: AM = $50.5$, GM = $10$. The gap can be huge.

What about $5$ and $5$? AM = $5$, GM = $5$ — equal, because the numbers are equal.

The General Statement

The same idea works for any number of non‑negative numbers. For $n$ numbers $x_1, x_2, \dots, x_n \ge 0$,

with equality if and only if all $x_i$ are equal.

What It Really Means …