Системи с параметър

[Гост]

|2x-a| < x+3

[ammornil]

[tex]|2x-a|<x+3 \\
\vspace{12} \\
\hspace{96}\fbox{DM \\
\forall x \in R} \\
\vspace{12} \\
1) x \in (-\infty; \frac{a}{2}) \Rightarrow 2x-a<0 \\
-(2x-a)<x+3 \\
-2x-x<3-a \\
3x>a-3 \\
x>\frac{a-3}{3} \\
\vspace{6} \\
\hspace{296} \frac{a-3}{3}-\frac{a}{2}<0 \\
\hspace{296} \frac{2a-6-3a}{6}=\frac{-a-6}{6}<0 \\
\hspace{296} -\frac{a}{6}-1<0 \Rightarrow a > -6\\
\vspace{6} \\
\hspace{96} 1.1.) a \in (0; -6] \Rightarrow x \in \oslash \\
\hspace{96} 1.2.) a \in (-6;+\infty) \Rightarrow x \in (\frac{a-3}{3};\frac{a}{2}) \\
\vspace{12} \\
2) x \in [\frac{a}{2}; +\infty) \Rightarrow 2x-a \ge 0 \\
2x-a<x+3 \\
2x-x<a+3 \\
x<a+3 \\
\vspace{6} \\
\hspace{296} a+3 -\frac{a}{2} > 0 \\
\hspace{296} 2a+6-a > 0 \Rightarrow 3a > -6 \Rightarrow a>-2\\
\vspace{6} \\
\hspace{96}2.1.) a \in (-\infty;-2] \Rightarrow x \in \oslash \\
\hspace{96}2.2.) a \in (-2; +\infty) \Rightarrow x \in [\frac{a}{2};a+3)[/tex]


Отговор:
[tex]a \in (-\infty;-6] \hspace{24} \Rightarrow x \in \oslash \\
a \in (-6;-2] \hspace{24} \Rightarrow x \in (\frac{a-3}{3};\frac{a}{2}) \\
a \in (-2; +\infty) \hspace{24} \Rightarrow x \in (\frac{a-3}{3};a+3)[/tex]