Number of solutions of the equation  $3^{\left|x\right|}=\left|2-\left|x\right|\right|$  is:

$3^{\left|x\right|}$ and $\left|2-\left|x\right|\right|$  both are even functions,

Consider the equation for $x\ge 0.$ i.e., $3^x=\left|2-x\right|$

Graph shows that there is only one root of the equation exists for $x\ge 0.$  

Therefore, we have two solutions in total.