Given #3(x-10)+2 >= 8#
We can subtract #2# from both sides without effecting the orientation of the inequality:
#color(white)("XXXX")##3abs(x-10)>=6#
Similarly dividing by a value #>0# does not effect the orientation of the inequality:
#color(white)("XXXX")##abs(x-10) >=2#
Case 1: #x<10#
#color(white)("XXXX")##(x-10)# is negative so we have
#color(white)("XXXX")##color(white)("XXXX")##10-x >= 2#
#color(white)("XXXX")#Subracting 10 from both sides:
#color(white)("XXXX")##color(white)("XXXX")##-x>=-8#
#color(white)("XXXX")#Multiplying by #(-1)# and remembering to reverse the direction of the inequality for a negative multiplier:
#color(white)("XXXX")##color(white)("XXXX")##x<=8#
Combining the two conditions #x<10# and #x<=8# for Case 1, we have
#color(white)("XXXX")##x<=8#
Case 2: #x>=10#
#color(white)("XXXX")##(x-10)# is positive so we have
#color(white)("XXXX")##color(white)("XXXX")##x-10>=2#
#color(white)("XXXX")#Adding 10 to both sides:
#color(white)("XXXX")##color(white)("XXXX")##x>=12#
Combining the two conditions #x>=10# and #x>=12# for Case 2, we have
#color(white)("XXXX")##x>=12#
Combining Case 1 with Case 2
(Note either Case 1 or Case 2 is possible)
#color(white)("XXXX")##x<=8# or #x>=12#
