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Difference between revisions of "Inequality"

(→‎Theorems: add carleman's by boy soprano)
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*If <math>a \ge b</math>, then <math>a+c>b</math>, where <math>c>0</math>.
 
*If <math>a \ge b</math>, then <math>a+c>b</math>, where <math>c>0</math>.
  
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== Theorems ==
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==Solving Inequalities==
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A common application of inequalities is solving them for a variable. For example, consider the inequality <math>5x+7>3x+8</math>. We can solve for the variable <math>x</math> here and get <math>2x-1>0\Leftrightarrow x>\frac{1}{2}</math>, thus placing implicit restrictions upon the variable <math>x</math>. A more complex example is <math>\frac{x-8}{x+5}+4\ge 3</math>. Here, we manipulate the right hand side: <math>\frac{x+5-13}{x+5}+4\ge 3 \Leftrightarrow 1-\frac{-13}{x+5}+4\ge 3 \Leftrightarrow x+5-13+4x+20\ge 3x+15\Leftrightarrow x\ge \frac{3}{2}</math>. This type of inequality is true only for <math>x</math> satisfying the restriction that the solver will need to find.
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==Complete Inequalities==
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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 <math>x</math>, <math>x^2\ge 0</math>. Most inequalities of this type hold only for positive numbers, and this type of inequality often has very clever problems and applications.
 +
 
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== List of Theorems ==
 
Here are some of the more useful inequality theorems, as well as general inequality topics.
 
Here are some of the more useful inequality theorems, as well as general inequality topics.
 
* [[Arithmetic Mean-Geometric Mean | Arithmetic Mean-Geometric Mean Inequality]]
 
* [[Arithmetic Mean-Geometric Mean | Arithmetic Mean-Geometric Mean Inequality]]

Revision as of 12:44, 5 January 2008

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 $a$ and $b$:

  • $a>b$ if $a$ is greater than $b$, that is, $a-b$ is positive.
  • $a<b$ if $a$ is smaller than $b$, that is, $a-b$ is negative.
  • $a\ge b$ if $a$ is greater than or equal to $b$, that is, $a-b$ is either positive or $0$.
  • $a\le b$ if $a$ is less than or equal to $b$, that is, $a-b$ is either negative or $0$.

Note that if and only if $a>b$, $b<a$, and vice versa. The same applies to the latter two signs: if and only if $a\ge b$, $b\le a$, and vice versa.

Some properties of inequalities are:

  • If $a>b$, then $a+c>b$, where $c\ge 0$.
  • If $a \ge b$, then $a+c\ge b$, where $c\ge 0$.
  • If $a \ge b$, then $a+c>b$, where $c>0$.

Solving Inequalities

A common application of inequalities is solving them for a variable. For example, consider the inequality $5x+7>3x+8$. We can solve for the variable $x$ here and get $2x-1>0\Leftrightarrow x>\frac{1}{2}$, thus placing implicit restrictions upon the variable $x$. A more complex example is $\frac{x-8}{x+5}+4\ge 3$. Here, we manipulate the right hand side: $\frac{x+5-13}{x+5}+4\ge 3 \Leftrightarrow 1-\frac{-13}{x+5}+4\ge 3 \Leftrightarrow x+5-13+4x+20\ge 3x+15\Leftrightarrow x\ge \frac{3}{2}$. This type of inequality is true only for $x$ satisfying the restriction that the solver will need to find.

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 $x$, $x^2\ge 0$. Most inequalities of this type hold only for positive numbers, and this type of inequality often has very clever problems and applications.

List of Theorems

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

Problems

Introductory

  • Given that $(a+1)(b+1)(c+1) = 8$, show that $abc \le 1$. (<url>weblog_entry.php?t=172070 Source</url>)

Intermediate

  • 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 $.500$. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than $.503$. What's the largest number of matches she could've won before the weekend began? (Source)

Olympiad

  • Let $a,b,c$ be positive real numbers. Prove that

$\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$ (Source)

Resources

Books

Intermediate

Olympiad

Articles

Olympiad


Classes

Olympiad


See also

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