Question

If $|x-2|+|x-3|=7,$ then the value of $x$ is

• A. $-1$
• B. 6
• C. $-1$ or 6
• D. None of these

Answer 1

Answer: (C)

Given, $|x-2|+|x-3|=7$ ...(i)
When $x<2,|x-2|=-(x-2)$ and $|x-3|=-(x-3)$
Then, from Eq. (i), $-(x-2)+\{-(x-3)\}=7$
$\Rightarrow$ $-x+2-x+3=7$
$\Rightarrow$ $-2x=7-5$
$\Rightarrow$ $2x=-2$
$\Rightarrow$ $x=-1<2$
When $2\le x<3,|x-2|=(x-2)$ and $|x-3|=-(x-3),$
Then, from Eq. (i), $x-2-x+3=7$
$\Rightarrow$ $1\ne 7$ So, the solution is not possible.
When $x\ge 3,|x-2|=(x-2)$ and $|x-3|=(x-3)$
Then, from Eq. (i), $x-2+x-3=7$
$\Rightarrow$ $2x=7+5$
$\Rightarrow$ $x=6>3$
Hence, $x=6$ or $-1$