How do you solve the inequality #-6abs(2-x)<-12#?

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Answer 1 · 2018-05-11T12:36:35

The inequality is fulfilled when
#x<0# or #x>4#, i.e. #x!in[0, 4]#

Let's draw the graph of this inequality:

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As we see on the graph x must be outside the area [0, 4] to solve this equation, i.e. #x<0# or #x>4#

To solve this matematically we need to remember that #|2-x|# means the absolut value of #(2-x)#, i.e.
#2-x# when #x<=2#
#x-2# when #x>=2#

We, therefore must consider two inequalities, one when #x<=2#, the other when #x>=2#

#x<=2#:
#-6(2-x)=-12+6x<-12# Add 12 on both sides:
#-12+6x+12<-12+12#
or #6x<0#. Therefore #x<0#

#x>=2#:
#-6(x-2)=-6x+12<-12# Add #6x+12# on both sides:
#-6x+12+6x+12<-12+6x+12#
#24<6x# If we divide on 6 on both sides, we end up with:
#x>4#

The inequality is fulfilled when
#x<0# or #x>4#, in other words #x!in[0, 4]#