How do you solve absolute value inequality #abs(2x-4)-1>0#?

1 Answer
Alan P.
Apr 7, 2015

Split the absolute value sub-expression into its two cases
#(2x-4)<0 rarr x<2#
and
#(2x-4)>=0 rarr x>=2#

If #(2x-4)<0#
then
#abs(2x-4)-1>0#
is equivalent to
#-2x+4-1 >0#
#-2x > -3#
#x < 2/3# remember multiplying by a negative reverses the inequality.
(Note that this is consistent with the requirement #x<2#)

If #(2x-4)>=0#
then
#abs(2x-4) -1>0#
is equivalent to
#2x-4-1 >=0#
#2x>5#
#x>5/2#
(Again, note that this is consistent with the requirement #x>=2#)

So the solution to the given inequality is all values of #x# such that
#x<2/3# or #x> 2 1/2#

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