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Proprietatile modulelor

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$$\hspace{5pt}Fie\hspace{3pt} x \in \mathbb{R}\hspace{3pt} un\hspace{3pt} numar\hspace{3pt} real,\hspace{3pt} oarecare.$$ $$\hspace{5pt}\underline{Definitie:}\hspace{3pt} Modulul\hspace{3pt} numarului\hspace{3pt} real\hspace{3pt} x\hspace{3pt} se\hspace{3pt} noteaza\hspace{3pt} |x|\hspace{3pt} si\hspace{3pt} reprezinta\hspace{3pt} distanta\hspace{3pt} de\hspace{3pt} la\hspace{3pt} origine\hspace{3pt} pana\hspace{3pt} la\hspace{3pt}$$ $$punctul\hspace{3pt} de\hspace{3pt} pe\hspace{3pt} axa\hspace{3pt} corespunzator\hspace{3pt} numarului\hspace{3pt} real\hspace{3pt} x.$$ $$\hspace{5pt}\underline{Proprietati:}$$
 $$1.\hspace{3pt} |x| \ge 0$$ $$2.\hspace{3pt} |x| = 0 \Leftrightarrow x = 0$$ $$3.\hspace{3pt} |-x| = |x|$$ $$4.\hspace{3pt} -|x| \le x \le |x|$$ $$5.\hspace{3pt} |x| = |y| \Leftrightarrow x = y\hspace{3pt} sau\hspace{3pt} x = -y$$ $$6.\hspace{3pt} |x| = a \Leftrightarrow x = \pm a,\hspace{3pt} a \in \mathbb{R}, a > 0$$ $$7.\hspace{3pt} |x| < a \Leftrightarrow x \in (-a;a) \Leftrightarrow -a < x < a$$ $$8.\hspace{3pt} |x| \le a \Leftrightarrow x \in [-a;a] \Leftrightarrow -a \le x \le a$$ $$9.\hspace{3pt} |x| > a \Leftrightarrow x \in (-\infty;-a)\cup(a;+\infty)$$ $$10.\hspace{3pt} |x| \ge a \Leftrightarrow x \in (-\infty;-a]\cup[a;+\infty)$$ $$11.\hspace{3pt} |x*y| = |x| * |y|$$ $$12.\hspace{3pt} \left|{x\over y}\right| = {{|x|}\over{|y|}}$$ $$13.\hspace{3pt} |x + y| \le |x| + |y|$$ $$14.\hspace{3pt} {|x|}^2 = (x)^2$$

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