# Trivial Inequality

The **trivial inequality** is an inequality that states that the square of any real number is nonnegative. Its name comes from its simplicity and straightforwardness.

## Contents

## Statement

For all real numbers , .

## Proof

We can have either , , or . If , then . If , then by the closure of the set of positive numbers under multiplication. Finally, if , then again by the closure of the set of positive numbers under multiplication.

Therefore, for all real , as claimed.

## Applications

The trivial inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.

One application is maximizing and minimizing quadratic functions. It gives an easy proof of the two-variable case of the Arithmetic Mean-Geometric Mean inequality:

Suppose that and are nonnegative reals. By the trivial inequality, we have , or . Adding to both sides, we get . Since both sides of the inequality are nonnegative, it is equivalent to , and thus we have as desired.

Another application will be to minimize/maximize quadratics. For example,

Then, we use trivial inequality to get if is positive and if is negative.

## Problems

### Introductory

- Find all integer solutions of the equation .
- Show that . Solution
- Show that for all real and .

### Intermediate

- Triangle has and . What is the largest area that this triangle can have? (AIME 1992)

- The fraction,

where and are side lengths of a triangle, lies in the interval , where and are rational numbers. Then, can be expressed as , where and are relatively prime positive integers. Find . (Solution here see problem 3 solution 1)

### Olympiad

- Let be the length of the hypotenuse of a right triangle whose two other sides have lengths and . Prove that . When does the equality hold? (1969 Canadian MO)

- Let and be real numbers. Show that

(Solution here see problem 13 solution 1)