Solve the absolute value of inequalities and express in interval notation for #abs(3(x+2)-7x) <=6#?

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Answer 1 · 2015-05-05T11:18:02

Simplify the expression within the absolute value then consider the two cases where that expression has a negative and a non-negative value.

#abs(3(x+2)-7x) <=6#

#abs(6-4x)<=6#

Case 1
If #6-4x<0#
then #abs(6-4x) = 4x-6#
and #x>3/2#

#4x-6<=6#
#4x<=12#
#x<=3#

So in this case #3/2<x<=3#

Case 2
If #6-4x>=0#
then #abs(6-4x) = 6-4x#
and #x<=3/2#

#6-4x <=6#
#-4x<=0#
#x>=0#

In this case #0<=x<=3/2#

Combining the cases
Valid solutions are values of #x# for which
either #3/2<x<=3#
or #0<=x<=3/2#

which combines as
#0<=x<=3#