The first sentence is not true in general. Consider the equation [tex]|x|=-1[/tex]. There are no solutions. Now consider [tex]|x|=0[/tex]. There is only one solution, [tex]x=0[/tex].
But whatever. You're asked to demonstrate that [tex]|x|=9[/tex] has two solutions (which is true; the right hand side must be a positive integer in order to have two solutions). This follows immediately from the definition of absolute value, which says
[tex]|x|=\begin{cases}x&\text{for }x\ge0\\-x&\text{for }x<0\end{cases}[/tex]
So suppose [tex]x\ge0[/tex]. Then
[tex]|x|=9\implies x=9[/tex]
Now suppose [tex]x<0[/tex]. Then
[tex]|x|=9\implies -x=9\implies x=-9[/tex]
So two solutions to [tex]|x|=9[/tex] are [tex]x=\pm9[/tex].
