absolute value inequalities

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:

$\begin{array}[]{ccc}a|f(x)|+b\leq c&&c\geq a|f(x)|+b\\ a|f(x)|+b<c&&c>a|f(x)|+b\\ a|f(x)|+b\geq c&&c\leq a|f(x)|+b\\ a|f(x)|+b>c&&c<a|f(x)|+b\end{array}$

Before using the mnemonic device, the expression $|f(x)|$ should be and on the left hand of the inequality. Once this is accomplished, the absolute value must be dealt with: One statement should look to the previous one, the only being that the absolute value are dropped. The other statement should also have the absolute value dropped, but the inequality needs reversed and the number (on the ) needs to be negated.

The two statements as described above should be using either $\operatorname{or}$ or $\operatorname{and}$ . The mnemonic that aids in remembering which one to use is greator less thand. That is, when the inequality before splitting up has $>$ or $\geq$ , the connector $\operatorname{or}$ should be used; when the inequality before splitting up has $<$ or $\leq$ , the connector $\operatorname{and}$ should be used.

Here is an example:

$\begin{array}[]{rl}8&>3+|2x-7|\\ 5&>|2x-7|\\ |2x-7|&<5\end{array}$

Since the inequality is $<$ , $\operatorname{and}$ should be used.

$\begin{array}[]{rlcrl}2x-7&<5&\,\operatorname{and}&2x-7&>-5\\ 2x&<12&\,\operatorname{and}&2x&>2\\ x&<6&\,\operatorname{and}&x&>1\end{array}$

1<x<6

I would like to thank Mrs. Sue Millikin, who taught me absolute value inequalities in this manner.

Title	absolute value inequalities
Canonical name	AbsoluteValueInequalities
Date of creation	2013-03-22 15:59:20
Last modified on	2013-03-22 15:59:20
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	7
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 97D40
Related topic	InequalityWithAbsoluteValues
Defines	greator less thand