|X - 1| + |x - 2| + |x - 3| ≥ 6  

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|x - 1| = (x - 1)  when x ≥ 1

         = - (x - )  when x < 1

|x - 2| = (x - 2)  when x ≥ 2

          = - (x - 2) when  x < 2

|x - 3| = (x -3) when  x ≥ 3

          = - (x - 3) when x < 3  

______________________

for   x < 1

|x - 1| + |x - 2| + |x - 3|  ≥ 6

= - (x - 1) - (x - 2) - (x - 3) ≥ 6  

= - 3x + 6 ≥ 6

add  ( - 6)  on both  sides

= > - 3x  ≥ 0

multiply by   ( - 1) both sides (equality will  be reversed) = > 3x ≤ 0 divide both sides by 3  

= > x  ≤ 0

- - - - - - - - - - - - - - - - - -

for  1 ≤ x < 2

|x - 1| + |x - 2| + |x - 3| ≥ 6

= (x - 1) - (x - 2) - (x - 3) ≥ 6

= - x + 4 ≥  6

subtract  4 on both sides

= > - x  ≥ 2

multiply  by ( - 1) both  sides (equality will  change)

= x ≤  - 2

- - - - - - - - - - - - - - - - - - - - -

for 2 ≤ x < 3

|x - 1| + |x - 2| + |x - 3| ≥  6

= (x - 1) + (x - 2) - (x - 3)  ≥ 6

= x ≥  6

- - - - - - - - - - - - - - - - - - - - - -

for x ≥ 3

|x - 1| + |x - 2| + |x - 3| >  6

= (x - 1) + (x - 2) + (x - 3) >  6

= 3x - 6  ≥ 6

add 6  on both sides

= > 3x ≥ 12 divide  by 3 on both sides

= > x  ≥ 4

- - - - - - - - - - - - - - - - - - - - - -

on  combining the  ranges we GET

x ≤ 0     or x ≥ 4