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 $