Inequality

The subject of mathematical inequalities is tied closely with optimization methods. While most of the subject of inequalities is often left out of the ordinary educational track, they are common in mathematics Olympiads.

Overview

Inequalities are arguably a branch of elementary algebra, and relate slightly to number theory. They deal with relations of variables denoted by four signs: $>,<,\ge,\le$ .

For two numbers $b$ :

$a-b$ is positive.
$a-b$ is negative.
$a-b$ is nonnegative.
$a-b$ is nonpositive.

Note that if and only if $b\le a$ , and vice versa.

Some properties of inequalities are:

If $c\ge 0$ .
If $c\ge 0$ .
If $c>0$ .

Solving Inequalities

In general, when solving inequalities, same quantities can be added or subtracted without changing the inequality sign, much like equations. However, when multiplying, dividing, or square rooting, we have to watch the sign. In particular, notice that although $-3 < -2$ . In particular, when multiplying or dividing by negative quantities, we have to flip the sign. Complications can arise when the value multiplied can have varying signs depending on the variable.

We also have to be careful about the boundaries of the solutions. In the example $x = \frac{3}{2}$ satisfies the inequality because the inequality is nonstrict.

Solutions can be written in interval notation. Closed bounds use square brackets, while open bounds (and bounds at infinity) use parentheses. For instance, $3 \le x < 6$ .

Linear Inequalities

Linear inequalities can be solved much like linear equations to get implicit restrictions upon a variable. However, when multiplying/dividing both sides by negative numbers, we have to flip the sign.

Polynomial Inequalities

The first part of solving polynomial inequalities is much like solving polynomial equations -- bringing all the terms to one side and finding the roots.

Afterward, we have to consider bounds. We're comparing the sign of the polynomial with different inputs, so we could imagine a rough graph of the polynomial and how it passes through zeroes (since passing through zeroes could change the sign). Then we can find the appropriate bounds of the inequality.

Rational Inequalities

A more complex example is $\frac{x-8}{x+5}+4\ge 3$ .

Here is a common mistake: $\begin{align*} \frac{x-8}{x+5}+4&\ge 3 \\\frac{x+5-13}{x+5}+4&\ge 3 \\1-\frac{13}{x+5}+4&\ge 3 \\x+5-13+4x+20&\ge 3x+15 \\x&\ge \frac{3}{2}. \end{align*}$ The problem here is that we multiplied by $x$ . Thus, we may have to reverse the direction of the inequality sign if we are multiplying by a negative number. But, we don't know if the quantity is negative either.

A correct solution would be to move everything to the left side of the inequality, and form a common denominator. Then, it will be simple to find the solutions to the inequality by considering the sign (negativeness or positiveness) of the fraction as $x$ varies. $\begin{align*} \frac{x-8}{x+5}+4 &\ge 3 \\ \frac{x-8}{x+5}+1 &\ge 0 \\ \frac{2x-3}{x+5} &\ge 0 \end{align*}$ We will start with an intuitive solution, and then a rule can be built for solving general fractional inequalities. To make things easier, we test positive integers. $3$ , and so on. All of these work. In fact, it's not difficult to see that the fraction will remain positive as $1$ , begin to cause a positive fraction? We can't just assume that $x$ . Solving the equation reveals that $x$ that is less than $\frac{3}{2}$ itself) seems to be a solution. Therefore, we conclude that the solutions are the intervals $(-\infty,-5)\cup[\frac{3}{2},+\infty)$ .

For the sake of better notation, define the "x-intercept" of a fractional inequality to be those values of $(-5,\frac{3}{2})$ , as well as one value in the region $[\frac{3}{2},+\infty)$ ; then we see which regions are part of the solution set. This does indeed give the complete solution set.

One must be careful about the boundaries of the solutions. In the example problem, the value $0$ .

Complete Inequalities

A inequality that is true for all real numbers or for all positive numbers (or even for all complex numbers) is sometimes called a complete inequality. An example for real numbers is the so-called Trivial Inequality, which states that for any real $x^2\ge 0$ . Most inequalities of this type are only for positive numbers, and this type of inequality often has extremely clever problems and applications.

List of Theorems

Here are some of the more useful inequality theorems, as well as general inequality topics.

Introductory

Advanced

Can someone fix that Ptolemy's is in Advanced?

Problems

Introductory

Practice Problems on Alcumus
- Inequalities (Prealgebra)
- Solving Linear Inequalities (Algebra)
- Quadratic Inequalities (Algebra)
- Basic Rational Function Equations and Inequalities (Intermediate Algebra)
A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $.503$ . What's the largest number of matches she could've won before the weekend began? (1992 AIME Problems/Problem 3)

Intermediate

Practice Problems on Alcumus
- Quadratic Inequalities (Algebra)
- Advanced Rational Function Equations and Inequalities (Intermediate Algebra)
- General Inequality Skills (Intermediate Algebra)
- Advanced Inequalities (Intermediate Algebra)
Given that $abc \le 1$ . (<url>weblog_entry.php?t=172070 Source</url>)

Olympiad

Let $\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$ (2001 IMO Problems/Problem 2)

Resources

Books

Intermediate

Olympiad

Advanced Olympiad Inequalities: Algebraic & Geometric Olympiad Inequalities by Alijadallah Belabess.
The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
Problem Solving Strategies by Arthur Engel contains significant material on inequalities.
Inequalities by G. H. Hardy, J. E. Littlewood, G. Pólya.

Articles

Olympiad

Inequalities by MIT Professor Kiran Kedlaya.
Inequalities by IMO gold medalist Thomas Mildorf.

Classes

Olympiad

The Worldwide Online Olympiad Training Program is designed to help students learn to tackle mathematical Olympiad problems in topics such as inequalities.

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Inequality

Contents

Overview

Solving Inequalities

Linear Inequalities

Polynomial Inequalities

Rational Inequalities

Complete Inequalities

List of Theorems

Introductory

Advanced

Problems

Introductory

Intermediate

Olympiad

Resources

Books

Intermediate

Olympiad

Articles

Olympiad

Classes

Olympiad

See also