Question

If x satisfies $\left(x-1\right|+\left(x-2\right|+\left(x-3\right|\ge 6$, then

• A. $0\le x\le 4$
• B. $x\le -2\mathrm{or}x\ge 4$
• C. $x\le 0\mathrm{or}x\ge 4$
• D. $-2\le x\le 4$

Answer 1

The correct option is C $x\le 0\mathrm{or}x\ge 4$

$\mathrm{Let}\left(x-1\right|+\left(x-2\right|+\left(x-3\right|<6\mathrm{or},\left(\left(x-1\right)+\left(x-2\right)+\left(x-3\right)\right|<\left(x-1\right|+\left(x-2\right|+\left(x-3\right|<6\mathrm{or},\left(3x-6\right|<6\mathrm{or},\left(x-2\right|<2\mathrm{or},-2<x-2<2\mathrm{or},0<x<4\mathrm{Hence},\mathrm{for}\left(x-1\right|+\left(x-2\right|+\left(x-3\right|\ge 6,x\le 0\mathrm{or}x\ge 4$