Question

Consider $|x-2e^{\ln 3}|+|x+2e^{\ln 3}|=12.$ Then which of the following statement(s) is(are) CORRECT ?

• A. The number of integral solutions of the equation is $7.$
• B. The number of integral solutions of the equation is $13.$
• C. The sum of all the integral solutions of the equation is $0.$
• D. The sum of all the integral solutions of the equation is $28.$

Answer 1

Answer: (B), (C)

The sum of all the integral solutions of the equation is $0.$

$|x-2e^{\ln 3}|+|x+2e^{\ln 3}|=12$
$\Rightarrow |x-6|+|x+6|=12$

Case$-1:x<-6$
$-x+6-x-6=12$
$\Rightarrow x=-6($ rejected $)$

Case$-2:-6\le x\le 6$
$-x+6+x+6=12$
$12=12$
$\therefore x\in [-6,6]$

Case$-3:x>6$
$x-6+x+6=12$
$\Rightarrow x=6($ rejected $)$
$\therefore x\in [-6,6]$

$\therefore$ Number of integral solutions is $13,\{-6,-5,...,5,6\}$ and sum of these integral solutions is zero.