The range of [tex]f(x) = |x|[/tex] is [tex]\left\{f(x) \in \Bbb R ~:~ f(x) \ge 0\right\}[/tex]. In other words, [tex]|x|[/tex] is non-negative, so [tex]|x|=-n[/tex] for any positive integer [tex]n[/tex] has no solution.
Just look at the definition of absolute value:
[tex]|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}[/tex]
If [tex]x\ge0[/tex], then [tex]|x|[/tex] doesn't change its value, since [tex]x[/tex] is already non-negative.
But if [tex]x<0[/tex], then [tex]|x|[/tex] negates the negative and returns a positive number [tex]-x[/tex].
