How do you solve # abs(2x - 6) = abs(x +7) - 3 #?

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Answer 1 · 2018-06-30T08:26:29

The solutions are #S={2/3,10}#

The equation is

#|2x-6|=|x+7|-3#

#=>#, #|2x-6|-|x+7|+3=0#

The points to be considered are when

#{(2x-6=0),(x+7=0):}#

#=>#, #{(x=3),(x=-7):}#

There are #3# cases to consider

In the interval #(-oo, -7)#

#2x-6-(-x-7)+3=0#

#=>#, #2x-6+x+7+3=0#

#=>#, #3x+4=0#

#=>#, #x=-4/3#

This solution is not possible since #x=-4/3# #!in# to #(-oo,-7)#

In the interval #(-oo, -7)#

#-2x+6-(x+7)+3=0#

#=>#, #-2x+6-x-7+3=0#

#=>#, #-3x+2=0#

#=>#, #x=2/3#

This solution is possible since #x=2/3# #in# to #(-7,3)#

In the interval #(3,+oo)#

#2x-6-(x+7)+3=0#

#=>#, #2x-6-x-7+3=0#

#=>#, #x-10=0#

#=>#, #x=10#

This solution is possible since #x=10# #in# to #(3,+oo)#

The solutions are #S={2/3,10}#

graph{|2x-6|-|x+7|+3 [-18.02, 18.03, -9.01, 9.01]}