Question

If the equation $|2-x|=|x+1|=k$ has exactly one solution, then number of integral values of $k$ is

• A. $7$
• B. $5$
• C. $4$
• D. $3$

Answer 1

The correct option is A $7$

Given $|2-x|-|x+1|=k$

C - I When $x-1$

$2-x+x+1=k$

$\therefore k=3$

C - II When $x\in [-1,2)$

$\therefore -2x\in (-4,2]$

$2-x-x-1=k$

$1-2x\in (-3,3]$

$=k=1-2x$

$\therefore k=-3,-2,-1,0,1,2,3$

C - III when $x\in [2,\infty )$

$x-2-x-1=k$

$k=-3$

Hence, Number of integral value of ${}^{\prime }{k}^{\prime }$ is ${}^{\prime }{7}^{\prime }$