Question

Number of solutions of the equation $\left|x-1\right|+\left|x-2\right|+\left|x-3\right|=k$ , $k>2$ is

Answer 1

Answer: 2

$\left|x-1\right|+\left|x-2\right|+\left|x-3\right|=k$
To find the number of solution we draw the graph of
$y=\left|x-1\right|+\left|x-2\right|+\left|x-3\right|$ and $y=k$
$y=|x-1|+|x-2|+|x-3|=\{\begin{array}{cc}-3x+6& x\le 1\\ 4-x& 1<x\le 2\\ x& 2<x\le 3\\ 3x-6& x>3\end{array}$
In the graph of above curve can be drawn as following


Clearly the number of solution of the equation
(i) for $k<2$ is zero.
(ii) for $k=2$ is only one
(iii) for $k>2$ , two solution are two