absolute value inequalities AbsoluteValueInequalities 2006-06-11 17:40:29 2006-08-20 01:05:30 Definition greator less thand % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $a,b,c \in \mathbb{R}$ and $f(x) \in \mathbb{R}[x]$. There is a mnemonic device that is useful for solving inequalities of the following forms: \begin{center} $\begin{array}{ccc} a|f(x)|+b \le c & \,\, & c \ge a|f(x)|+b \\ a|f(x)|+b < c & \,\, & c > a|f(x)|+b \\ a|f(x)|+b \ge c & \,\, & c \le a|f(x)|+b \\ a|f(x)|+b > c & \,\, & c < a|f(x)|+b \end{array}$ \end{center} Before using the mnemonic device, the expression $|f(x)|$ should be \PMlinkescapetext{isolated} and on the left hand \PMlinkescapetext{side} of the inequality. Once this is accomplished, the absolute value must be dealt with: One statement should look \PMlinkescapetext{similar} to the previous one, the only \PMlinkescapetext{difference} being that the absolute value \PMlinkescapetext{signs} are dropped. The other statement should also have the absolute value \PMlinkescapetext{signs} dropped, but the inequality needs reversed and the number (on the \PMlinkescapetext{right}) needs to be negated. The two statements as described above should be \PMlinkescapetext{connected} using either $\operatorname{or}$ or $\operatorname{and}$. The mnemonic that aids in remembering which one to use is {\sl greator less thand\/}. That is, when the inequality before splitting up has $>$ or $\ge$, the connector $\operatorname{or}$ should be used; when the inequality before splitting up has $<$ or $\le$, the connector $\operatorname{and}$ should be used. Here is an example: \begin{center} $\begin{array}{rl} 8 & > 3+|2x-7| \\ 5 & > |2x-7| \\ |2x-7| & < 5 \end{array}$ \end{center} Since the inequality is $<$, $\operatorname{and}$ should be used. \begin{center} $\begin{array}{rlcrl} 2x-7 & < 5 & \, \operatorname{and} \, & 2x-7 & > -5 \\ 2x & < 12 & \, \operatorname{and} \, & 2x & > 2 \\ x & < 6 & \, \operatorname{and} \, & x & > 1 \end{array}$ \end{center} $$1<x<6$$ I would like to thank Mrs. Sue Millikin, who taught me absolute value inequalities in this manner.