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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:
$\begin{array}{ccc} a|f(x)|+b \le c & \,\, & c \ge a|f(x)|+b \\ a|f(x)|+b < c & \,\, & c > a|f(x)|+b \\ a|f(x)|+b \ge c & \,\, & c \le a|f(x)|+b \\ a|f(x)|+b > c & \,\, & c < a|f(x)|+b \end{array}$
Before using the mnemonic device, the expression $|f(x)|$ should be isolated and on the left hand side of the inequality. Once this is accomplished, the absolute value must be dealt with: One statement should look similar to the previous one, the only difference being that the absolute value signs are dropped. The other statement should also have the absolute value signs dropped, but the inequality needs reversed and the number (on the right) needs to be negated.
The two statements as described above should be connected 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 $\ge$ 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:
$\begin{array}{rl} 8 & > 3+|2x-7| \\ 5 & > |2x-7| \\ |2x-7| & < 5 \end{array}$
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.
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