Question

Solve for $x:$

$|x-1|+|x-2|+|x-3|\ge 6$

Answer 1

This question contains a modulus operator so we have to do in two steps. Initially eliminate the modulus sign by giving positive to each term and solve. Then eliminate the modulus sign by giving negative to each term. This is the general procedure to questions with a modulus operator.

$|x-1|+|x-2|+|x-3|\ge 6$

Step 1: $+(x-1+x-2+x-3)\ge 6$

$\Rightarrow 3x-6\ge 6$

$\Rightarrow 3x\ge 12$

$\Rightarrow x\ge 4$

Step 2: $-(x-1+x-2+x-3)\ge 6$

$\Rightarrow -(3x-6)\ge 6$

$\Rightarrow -3x+6\ge 6$

$\Rightarrow -3x\ge 0$

$\Rightarrow -x\ge 0$

$\Rightarrow x\le 0$

therefore our solution is in two intervals. So we take the union of those intervals. Thus answer is $(-\infty ,0]\cup [4,\infty )$, and we can say that $x$ will be less than equal to zero and $x$ will be greater than equal to $4.$