The solution set of the inequation `|x-1|+|x-2|+|x-3|>= 6` is
The solution set of the inequation `|x-1|+|x-2|+|x-3|>= 6` is
A. `[0, 4]`
B. `(-oo, -2) cup [4, oo)`
C. `(-oo, 0]cup[4, oo)`
D. none of these
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Consider 4 cases:
Case 1: x<1
(1-x)+(2-x)+(3-x)>=6; 6-3x>=6; 3x<=0; x<=0 (since last is always <1, all x<=0 all solutions)
Case 2: 1<=x<2
(x-1)+(2-x)+(3-x)>=6; 4-x>=6; x<=-2 (no solutions here)
Case 3: 2<=x<3
(x-1)+(x-2)+(3-x)>=6; x>=6 (again, no solutions here)
Case 4: x>=3
(x-1)+(x-2)+(x-3)>=6; 3x-6>=6; 3x>=12; x>=4 (again, all x>=4 are solutions)
Summary:
The solutions are: ( − ∞ , 0 ] ∪ [ 4 , ∞ )
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