Q. Number of solutions of the equation $\left|x - 1\right|+\left|x - 2\right|+\left|x - 3\right|=k$ , $k>2$ is
NTA AbhyasNTA Abhyas 2022
Solution:
$\left|x - 1\right|+\left|x - 2\right|+\left|x - 3\right|=k$
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 $y=k$
$y=|x-1|+|x-2|+|x-3|=\left\{\begin{array}{cc}-3 x+6 & x \leq 1 \\ 4-x & 13\end{array}\right.$
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
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