# 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)|+ba|f(x)|+b\\ a|f(x)|+b\geq c&&c\leq a|f(x)|+b\\ a|f(x)|+b>c&&c

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

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

Title absolute value inequalities AbsoluteValueInequalities 2013-03-22 15:59:20 2013-03-22 15:59:20 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Definition msc 97D40 InequalityWithAbsoluteValues greator less thand