I 5-I 2-3xII=x-1 тема модулі

[latex]|5-|2-3x|\, |=x-1\quad \Rightarrow \\\\5-|2-3x|=\pm (x-1)\\\\a)\; \; 5-|2-3x|=-(x-1)\\\\|2-3x|=5+(x-1)[/latex] [latex]|2-3x|=x+4\; \; \to \; \; \left [ {{2-3x=x+4,\; x \leq \frac{2}{3}} \atop {2-3x=-(x+4),\; x>\frac{2}{3}}} \right. \; \left [ {{4x=-2,\; x \leq \frac{2}{3}} \atop {2x=6,\; x>\frac{2}{3}}} \right. [/latex]  [latex]\left [ {{4x=-2,\; x \leq } \atop {2x=6,\; x>\frac{2}{3}}} \right. \left [ {{x=-\frac{1}{2}} \atop {x=3}} \right. \\\\b)\; \; 5-|2-3x|=x-1\\\\|2-3x|=6-x\; \; \to \; \; \left [ {{2-3x=6-x,\; x \leq \frac{2}{3}} \atop {2-3x=-(6-x),\; x>\frac{2}{3}}} \right. \; \left [ {{2x=-4} \atop {4x=8}} \right. \; \left \{ {{x=-2} \atop {x=2}} \right. [/latex] [latex]b)\; \; 5-|2-3x|=x-1\\\\|2-3x|=6-x\; \; \to \; \; \left [ {{2-3x=6-x,\; x \leq \frac{2}{3}} \atop {2-3x=-(6-x),\; x>\frac{2}{3}}} \right. \; \left [ {{2x=-4} \atop {4x=8}} \right. \; \left \{ {{x=-2} \atop {x=2}} \right. \\\\Proverka:\; 1)\; x=-\frac{1}{2}:\; |5-|2+3\cdot \frac{1}{2}|\, |=|5-\frac{7}{2}|=\frac{3}{2}\; ;\\\\x-1=-\frac{1}{2}-1=-\frac{1}{2}-1=-\frac{3}{2}\; ;\\\\\frac{3}{2}\ne -\frac{3}{2}\\\\2)\; \; x=3:\; \; |5-|2-3\cdot 3||=|5-7|=|-2|=2\; ;\\\\x-1=3-1=2\; ;\\\\2=2[/latex] [latex]3)\; \; x=-2:\; \; |5-|2+3\cdot 2||=|5-8|=|-3|=3\; ;\\\\x-1=-2-1=-3\; ;\\\\3\ne -3\\\\4)\; \; x=2:\; \; |5-|2-3\cdot 2||=|5-4|=1\\\\x-1=2-1=1\\\\1=1\\\\Otvet:\; \; x=3\; ,\; \; x=2\; .[/latex]