Алгебра

[Гост]

x+1/x-1 + (4/ [tex]x^{2 }[/tex] + 2x - 3) - x+2/x+3

[ammornil]

$ \dfrac{x+1}{x-1}+\dfrac{4}{x^{2}+2x-3} +\dfrac{x+2}{x+3}\\[6pt] x^{2}+2x-3= x^{2} -x +3x -3= x(x-1) +3(x-1)= (x+3)(x-1) \\[6pt] \dfrac{x+1}{x-1}+\dfrac{4}{x^{2}+2x-3} +\dfrac{x+2}{x+3}= \dfrac{x+1}{x-1}+\dfrac{4}{(x+3)(x-1)} +\dfrac{x+2}{x+3} \\[6pt] \quad \text{ДМ}: \quad \begin{array}{|l} x+3\ne{0} \\ x-1\ne{0} \end{array} \Rightarrow \begin{array}{|l} x\ne{-3} \\ x\ne{1} \end{array} \Rightarrow x\in{}(-\infty;-3)\cup{}(-3;1)\cup{}(1;+\infty) \\[12pt] \underbrace{\dfrac{\overset{x+3}{x+1}}{x-1}+\dfrac{\overset{1}{4}}{(x+3)(x-1)} +\dfrac{\overset{x-1}{x+2}}{x+3}}_{(x+3)(x-1)}= \dfrac{(x+3)(x+1)+4\cdot{1}+(x-1)(x+2)}{(x+3)(x-1)}= \\[6pt] =\dfrac{x^{2}+x+3x+3+4+x^{2}+2x-x-2}{(x+3)(x-1)}= \dfrac{2x^{2}+5x+5}{x^{2}+2x-3} $

[Гост]

Благодаря!

[ammornil]

Грешно съм преписал един знак$\\[12pt] \dfrac{x+1}{x-1}+\dfrac{4}{x^{2}+2x-3} \boxed{\red{ \ -} }\dfrac{x+2}{x+3}\\[6pt] x^{2}+2x-3= x^{2} -x +3x -3= x(x-1) +3(x-1)= (x+3)(x-1) \\[6pt] \dfrac{x+1}{x-1}+\dfrac{4}{x^{2}+2x-3} -\dfrac{x+2}{x+3}= \dfrac{x+1}{x-1}+\dfrac{4}{(x+3)(x-1)} -\dfrac{x+2}{x+3} \\[6pt] \quad \text{ДМ}: \quad \begin{array}{|l} x+3\ne{0} \\ x-1\ne{0} \end{array} \Rightarrow \begin{array}{|l} x\ne{-3} \\ x\ne{1} \end{array} \Rightarrow x\in{}(-\infty;-3)\cup{}(-3;1)\cup{}(1;+\infty) \\[12pt] \underbrace{\dfrac{\overset{x+3}{x+1}}{x-1}+\dfrac{\overset{1}{4}}{(x+3)(x-1)} -\dfrac{\overset{x-1}{x+2}}{x+3}}_{(x+3)(x-1)}= \dfrac{(x+3)(x+1)+4\cdot{1}-(x-1)(x+2)}{(x+3)(x-1)}= \\[6pt] =\dfrac{x^{2}+x+3x+3+4-(x^{2}+2x-x-2)}{(x+3)(x-1)}= \dfrac{\cancel{x^{2}}+4x+7\cancel{-x^{2}} -x +2}{(x+3)(x-1)}= \\[6pt] = \dfrac{3x+9}{(x+3)(x-1)}= \dfrac{3\cancel{(x+3)}}{\cancel{(x+3)}(x-1)}= \dfrac{3}{x-1} $