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

$ 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 $
$-x-x+2=2 \Rightarrow-2 x=0 \Rightarrow x=0$(No solution)
$\text { Case-II : } 0 \leq x \leq 2$
$\Rightarrow x-x+2=2$
$\Rightarrow 2=2$
$\therefore x \in[0,2]$
Case-III : $x>2$
$x+x-2=2 \Rightarrow 2 x=4 \Rightarrow x=2 \text { (no solution) }$
$\therefore x \in[0,2]$ is the solution set of the given equation
$\therefore $ Sum of all integers $=0+1+2=3$.