# Are all absolute value inequalities going to turn into compound inequalities?

Feb 10, 2015

If all the terms of the inequality are absolute values, then Yes, but if the inequality contains a term which is not force to an absolute value then No.

If all terms of an inequality are absolute values then the only way either side could be negative is if the collection of terms on that side contained a subtraction. For example:
$| a | - | b | < | c |$
But such an inequality could always be written without the subtraction by adding an amount equal to that being subtracted to each side
$\left(| a | - | b | < | c |\right) \to \left(| a | < | c | + | b |\right)$

Since this is the case one side of the inequality must have an absolute value that is less than (or less than or equal to) some positive number. Lets call this positive number $K$.

The minimal side of the inequality must be $< \left(K\right)$ AND $> \left(- K\right)$ (a compound relationship).

Note however, if any of the terms are not absolute values, this does not apply. For example:
$a < | b |$
is not a compound relationship; $a$ is not restricted except by an upper limit of $| b |$.