Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 15: More Inequalities

Lesson 15.3: The Trivial Inequality

In this Grade 4 AMC Math lesson from AoPS Introduction to Algebra, students explore the Trivial Inequality, which states that the square of any real number is always greater than or equal to zero. Students learn to apply this fundamental principle to prove related inequalities, including the AM-GM inequality for two variables, using the algebraic technique of working backwards from a desired result. The lesson also emphasizes why valid proofs must be written forwards, starting from known true statements.

Section 1

Trivial Quadratic Inequalities

Property

A quadratic inequality is called trivial when it is either always true for all real numbers or never true (no solution).

For a quadratic expression $ax^2 + bx + c$ :

If $a > 0$ and the parabola has no real roots, then $ax^2 + bx + c > 0$ for all real numbers
If $a < 0$ and the parabola has no real roots, then $ax^2 + bx + c < 0$ for all real numbers

Section 2

Non-negative Nature of Squares

Property

For any real number $x$ , we have $x^2 \geq 0$ . This means that the square of any real number is always non-negative.

Examples

Section 3

Deriving AM-GM from the Trivial Inequality

Property

The AM-GM inequality states that for nonnegative real numbers $a$ and $b$ :

\frac{a + b}{2} \geq \sqrt{ab}

This can be derived from the trivial inequality $(\sqrt{a} - \sqrt{b})^2 \geq 0$ .

Section 4

Standard Inequality Proofs Using the Trivial Inequality

Property

Key inequalities proven from $x^2 \geq 0$ :

\frac{a^2 + b^2}{2} \geq ab \text{ for all real numbers } a, b

x + \frac{1}{x} \geq 2 \text{ for all positive real numbers } x

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Trivial Quadratic Inequalities

Property

A quadratic inequality is called trivial when it is either always true for all real numbers or never true (no solution).

For a quadratic expression $ax^2 + bx + c$ :

If $a > 0$ and the parabola has no real roots, then $ax^2 + bx + c > 0$ for all real numbers
If $a < 0$ and the parabola has no real roots, then $ax^2 + bx + c < 0$ for all real numbers

Section 2

Non-negative Nature of Squares

Property

For any real number $x$ , we have $x^2 \geq 0$ . This means that the square of any real number is always non-negative.

Examples

Section 3

Deriving AM-GM from the Trivial Inequality

Property

The AM-GM inequality states that for nonnegative real numbers $a$ and $b$ :

\frac{a + b}{2} \geq \sqrt{ab}

This can be derived from the trivial inequality $(\sqrt{a} - \sqrt{b})^2 \geq 0$ .

Section 4

Standard Inequality Proofs Using the Trivial Inequality

Property

Key inequalities proven from $x^2 \geq 0$ :

\frac{a^2 + b^2}{2} \geq ab \text{ for all real numbers } a, b

x + \frac{1}{x} \geq 2 \text{ for all positive real numbers } x

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Trivial Quadratic Inequalities

Property

A quadratic inequality is called trivial when it is either always true for all real numbers or never true (no solution).

For a quadratic expression $ax^2 + bx + c$ :

If $a > 0$ and the parabola has no real roots, then $ax^2 + bx + c > 0$ for all real numbers
If $a < 0$ and the parabola has no real roots, then $ax^2 + bx + c < 0$ for all real numbers

Section 2

Non-negative Nature of Squares

Property

For any real number $x$ , we have $x^2 \geq 0$ . This means that the square of any real number is always non-negative.

Examples

Section 3

Deriving AM-GM from the Trivial Inequality

Property

The AM-GM inequality states that for nonnegative real numbers $a$ and $b$ :

\frac{a + b}{2} \geq \sqrt{ab}

This can be derived from the trivial inequality $(\sqrt{a} - \sqrt{b})^2 \geq 0$ .

Section 4

Standard Inequality Proofs Using the Trivial Inequality

Property

Key inequalities proven from $x^2 \geq 0$ :

\frac{a^2 + b^2}{2} \geq ab \text{ for all real numbers } a, b

x + \frac{1}{x} \geq 2 \text{ for all positive real numbers } x