Valor absoluto
$$
\left\{\begin{array}{c}
|x| = \left\{\begin{array}{ccc}
x & si & x \geq 0 \\
-x & si & x < 0 \\
\end{array}\right.\\
\\
|x| \geq 0 \\
\\
|x| \leq a \Leftrightarrow -a \leq x \leq +a \Leftrightarrow x \in [-a,+a]\\
\\
|x| < a \Leftrightarrow -a < x < +a \Leftrightarrow x \in (-a,+a)\\
\\
|x| \geq a \Leftrightarrow
\left\{\begin{array}{c}
x \leq -a \\
\vee \\
x \geq +a \\
\end{array}\right. \Leftrightarrow x \in (-\infty,-a] \cup [+a,+\infty)\\
\\
|x| > a \Leftrightarrow
\left\{\begin{array}{c}
x < -a \\
\vee \\
x > +a \\
\end{array}\right. \Leftrightarrow x \in (-\infty,-a) \cup (+a,+\infty)\\
\\
|x-b| \leq r \Leftrightarrow b-r \leq x \leq b+r \Leftrightarrow x \in [b-r,b+r] \Leftrightarrow x \in E[b,r] \\
\\
|x-b| < r \Leftrightarrow b-r < x < b+r \Leftrightarrow x \in (b-r,b+r) \Leftrightarrow x \in E(b,r)\\
\\
|x-b| \geq r \Leftrightarrow
\left\{\begin{array}{c}
x \leq b-r \\
\vee \\
x \geq b+r \\
\end{array}\right. \Leftrightarrow x \in (-\infty,b-r] \cup [b+r,+\infty)\\
\\
|x-b| > r \Leftrightarrow
\left\{\begin{array}{c}
x < b-r \\
\vee \\
x > b+r \\
\end{array}\right. \Leftrightarrow x \in (-\infty,b-r) \cup (b+r,+\infty)\\
\\
\end{array}\right.
$$
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