Solve |2x-3|+|x-1|=|x-2| Find the values of x?

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Answer 1 · 2018-06-24T08:08:09

The solutions are #S={1, 3/2}#

The equation is

#|2x-3|+|x-1|=|x-2|#

There are #3# points to consider

#{(2x-3=0),(x-1=0),(x-2=0):}#

#=>#, #{(x=3/2),(x=1),(x=2):}#

There are #4# intervals to consider

#{(-oo,1),(1,3/2),(3/2,2),(2,+oo):}#

On the first interval #(-oo,1)#

#-2x+3-x+1=-x+2#

#=>#, #2x=2#

#=>#, #x=1#

#x# fits in this interval and the solution is valid

On the second interval #(1, 3/2)#

#-2x+3+x-1=-x+2#

#=>#, #0=0#

There is no solution in this interval

On the third interval #( 3/2,2)#

#2x-3+x-1=-x+2#

#=>#, #4x=6#

#=>#, #x=6/4=3/2#

#x# fits in this interval and the solution is valid

On the fourth interval #(2, +oo)#

#2x-3+x-1=x-2#

#=>#, #2x=2#

#=>#, #x=1#

#x# does not fit in this interval.

The solutions are #S={1, 3/2}#