Question

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

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

Answer 1

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

Let $|x-1|+|x-2|+|x-3|<6$
$\Rightarrow |(x-1)+(x-2)+(x-3)|<|x-1|+|x-2|+|x-3|<6$
$\Rightarrow |3x-6|<6$
​​​​​​​$\Rightarrow |x-2|<2$
​​​​​​​​​​​​​​$\Rightarrow -2<x-2<2$
​​​​​​​​​​​​​​$\Rightarrow 0<x<4$

Hence, for $|x-1|+|x-2|+|x-3|\ge 6$,
we should have $x\le 0$ or $x\ge 4$