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. none of these

Answer 1

The correct option is C

$x\le 0$ or $x\ge 4$

We have to solve
$|x-1|+|x-2|+|x-3|\ge 6$
Let $|x-1|+|x-2|+|x-3|<6$
Now $|(x-1)+(x-2)+(x-3)|<|x-1|+|x-2|+|x-3|<6$
$\therefore |3x-6|<6\therefore |x-2|<2\therefore -2<x-2<2$
or $0<x<4$
Hence, for $|x-1|+|x-2|+|x-3|\ge 6,x\le 0$ or $x\ge 4$

Alternative Method:
We have f(x) =|x-1|+|x-2|+|x-3|
To draw the graph of the function, we consider
f(0) =1 +2 +3 =6
f(1) =0 +1 +2 =3
f(2) =1+0+1 =2
f(3) =2 +1 +0 =3
f(4) =3 +2 +1 =6
Now, join these points with line segments as shown in the following figure.
Note: The methods to drew the graphs of such functions is explained in the book "Graphs for JEE Main and JEE Advanced".
Also, draw the line y =6

For $f\left(x\right)=|x-1|+|x-2|+|x-3|\ge 6$, graph $y=f\left(x\right)$ must lie above the graph of $y=6$.
As shown in the figure it occurs for $x\le 0$ or $x\ge 4$