If x satisfies `|x-1|+x-2|+|x-3| >= 6,` then (a) `0 le x le 4` (b) `x le 4` (c) `x le 0 or x >= 4` (d) none of these

Explanation

Case I) `x le1`, then
`1-x+2-x+3-x ge6 rArr xle0`
Hence, `xle0` …………..(i)
Case II) `1 lt x le2`, then
`x-1+2-x+3-x ge6 rArr xle-2`
But `1 lt x le 2 rArr` No solution. …………..(ii)
Case III) `2 lt x le3`, then
`x-1+x-2+3 -x ge6 rArr xge6`
But `2 lt x le3 rArr` No solution..............(iii)
Case IV: `x lt 3`, then
`x-1+x-2 =x-3 ge6 rArr xge4`
Hence, `xge4` ................(iv)
From (i), (ii), (iii) and (iv) the given inequality holds for `x le0` or `xge4`