Linear Inequalities: Variations
Linear inequalities with absolute values
As in the chapter Linear equations with one unknown, inequalities with absolute values can be solved by reducing them into linear inequalities.
The absolute value of #x# is the distance from #x# to #0# on the number line and is noted as #|x|# i.e.: \[|x|= \begin{cases}x&\text{ als } x\ge0\\ -x&\text{ als } x\lt0 \end {cases}\]
Therefore #|-7| = 7# and #|3|=3# .
Let #a,b,c# be real numbers. Inequality #|ax+b|\ge c# is equivalent to
\[\left(ax+b\ge 0 \land ax+b\ge c\right)\lor\left(ax+b\le 0 \land ax+b\le -c\right)\tiny.\]
#\displaystyle (x\ge {{8}\over{7}}) \vee (x\le 0)#
Using the definition of absolute value, we can rephrase the inequality as #(7x-4\ge 0 \wedge 7x-4\ge 4) \vee (7x-4\lt 0 \wedge -7x+4\ge 4)#. Simplifying the inequalities involved in this expression using the Rules of calculation for inequalities, we find it to be equivalent to #\displaystyle (x\ge {{4}\over{7}} \wedge x\ge {{8}\over{7}}) \vee (x\lt {{4}\over{7}} \wedge x\le 0)#. As #\displaystyle {{4}\over{7}}\lt {{8}\over{7}}# and #\displaystyle {{4}\over{7}} \gt 0#, we conclude that the solution is #\displaystyle (x\ge {{8}\over{7}}) \vee (x\le 0)#.
Using the definition of absolute value, we can rephrase the inequality as #(7x-4\ge 0 \wedge 7x-4\ge 4) \vee (7x-4\lt 0 \wedge -7x+4\ge 4)#. Simplifying the inequalities involved in this expression using the Rules of calculation for inequalities, we find it to be equivalent to #\displaystyle (x\ge {{4}\over{7}} \wedge x\ge {{8}\over{7}}) \vee (x\lt {{4}\over{7}} \wedge x\le 0)#. As #\displaystyle {{4}\over{7}}\lt {{8}\over{7}}# and #\displaystyle {{4}\over{7}} \gt 0#, we conclude that the solution is #\displaystyle (x\ge {{8}\over{7}}) \vee (x\le 0)#.
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