[tex]a) $ $ \big{|} |x-3|-2\big{|} = 1 \Rightarrow |x-3|-2 = \pm 1 \\ \\ \fbox{1}$ $ |x-3|-2 = 1 \Rightarrow |x-3|=3 \Rightarrow x-3 = \pm 3 \\ \\ \\ \fbox{1'} $ $ x-3 = 3 \Rightarrow \boxed{x = 6} \\ \\ \fbox {1''} $ $ x-3 = -3 \Rightarrow \boxed{x = 0} \\ \\ \\ \fbox{2}$ $ |x-3| - 2 = -1 \Rightarrow x-3| = 1 \Rightarrow x-3 = \pm 1 \\ \\ \\ \fbox{2'} $ $ x-3 = 1 \Rightarrow \boxed{x = 4} \\ \\ \fbox{2''} $ $ x-3 =-1 \Rightarrow \boxed{x = 2} \\ \\ \Rightarrow S=\{0,2,4,6\}[/tex]
[tex]b) $ $ |3x+1|=3 \Rightarrow 3x+1 =\pm 3 \\ \\ \fbox{1} $ $ 3x+1 = 3\Rightarrow 3x = 2 \Rightarrow \boxed{x=\frac{2}{3} } \notin \mathbb{Z} $ \\ \\ \fbox{2}$ $ $3x+1 = -3 \Rightarrow 3x = -2 \Rightarrow \boxed{x= -\frac{2}{3} } \notin \mathbb{Z} \\ \\ \Rightarrow S = \O[/tex]
c)----
