Question

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

• A. $0\le x\le 4$
• B. $x\le -2orx\ge 4$
• C. $x\le 0orx\ge 4$
• D. $x\le -1orx\ge 5$

Answer 1

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

For $x\le 1,$ the given in equation becomes
$1-x+2-x+3-x\ge 6\Rightarrow -3x\ge 0\Rightarrow x\le 0$
and for $x\ge 3,$ the given equation becomes
$x-1+x-3+x-3\ge 6\Rightarrow 3x\ge 12\Rightarrow x\ge 4$
For $1<x\le 2$ we get
$x-1+2-x+3-x\ge 6\Rightarrow -x+4\ge 6$
$i.e.,-x\ge 2\Rightarrow x\le -2$ not possible.
For $2<x<3,$ we get $x-1+x-2+3-x\ge 6$
$\Rightarrow x\ge 6,$ not possible
hence solution set is
$(-\infty ,0]\cup [4,\infty ),i.e.x\le 0orx\ge 4$