Question

Find the sum of all the integral solution(s) of the equation $3^{|x|}={\left(\frac{3}{(\sqrt{3}{)}^{|x-2|}}\right)}^2$.

Answer 1

Answer: 3

$3^{|x|}={\left(\frac{3}{(\sqrt{3}{)}^{|x-2|}}\right)}^2\Rightarrow 3^{|x|}=\frac{9}{3^{|x-2|}}$
$\Rightarrow 3^{|x|}=3^{2-|x-2|}\Rightarrow |x|=2-|x-2|$
$\Rightarrow |x|+|x-2|=2$
$\text{ Case-I : }x<0$
$-x-x+2=2\Rightarrow -2x=0\Rightarrow x=0$(No solution)
$\text{ Case-II : }0\le x\le 2$
$\Rightarrow x-x+2=2$
$\Rightarrow 2=2$
$\therefore x\in [0,2]$
Case-III : $x>2$
$x+x-2=2\Rightarrow 2x=4\Rightarrow x=2\text{ (no solution) }$
$\therefore x\in [0,2]$ is the solution set of the given equation
$\therefore$ Sum of all integers $=0+1+2=3$.