Question

3x+2|x-1|=|x+2|
чему равно х? ​

Answer 1

Ответ:    $x\in [-2,1)\cup \{1\frac{1}{3}\}\; .$

Пошаговое объяснение:

$3x+2|x-1|=|x+2|\; \; \; \Rightarrow \; \; \; 3x+2|x-1|-|x+2|=0\\\\x-1=0\; \; \to \; \; x_1=1\\\\x+2=0\; \; \to \; \; x_2=-2\\\\---(-2)---(1)---\\\\\\a)\; \; x\leq -2:\; \; (x-1)<0\; \; \to \; \; |x-1|=-(x-1)=-x+1\; ;\\\\(x+2)<0\; \; \to \; \; |x+2|=-(x+2)=-x-2\; ;\\\\3x+2|x-1|-|x+2|=3x+2(-x+1)-(-x-2)=2x+4\; ,\\\\2x+4=0\; \; \to \; \; \underline {x=-2}\\\\\\b)\; \; -2<x\leq 1\, :\; \; (x-1)<0\ ;\; \to \; \; |x-1|=-x+1\; ;\; \\\\(x+2)>0\; \; \to \; \; |x+2|=x+2\; ;\\\\3x+2|x-1|-|x+2|=3x+2(-x+1)-(x+2)=0\; ;\\\\0=0\; \; verno\; \; \Rightarrow \; \; \underline {\; x\in (-2,\, 1\, ]\; }$

$c)\; \; x>1\, :\; \; (x-1)>0\; \; \to \; \; |x-1|=x-1\; ;\\\\(x+2)>0\; \; \to \; \; |x+2|=x+2\; ;\\\\3x+2|x-1|-|x+2|=2x+2(x-1)-(x+2)=3x-4\; ;\\\\\3x-4=0\; \; \to \; \; 3x=4\; ,\; \; x=\frac{4}{3}\; ,\; \; \underline {x=1\frac{1}{3}\in (1,+\infty )}\\\\\\Otvet:\; \; x\in \{-2\}\cup (-2,1\, ]\cup \{1\frac{1}{3}\}=[-2,1)\cup \{1\frac{1}{3}\} \; .$