Odpowiedź:
[tex]a)~~\dfrac{x^{2} -6x}{x+3} \div \dfrac{x}{x-2} =\dfrac{x^{2} -4x-12}{x+3}~~,~~D=R-\{ -3,-2,0\}[/tex]
[tex]b)~~\dfrac{x}{x^{2} +5x} \cdot \dfrac{x^{2} -25}{x+2} =\dfrac{x-5}{x+2}~~,~~D=R-\{-5,-2,0\}[/tex]
[tex]c)~~\dfrac{x-2}{x-1} +\dfrac{x-3}{x+2} =\dfrac{ 2x^{2} -4x-1}{ x^{2} +x-2}~~,~~D=R-\{-2,1\}[/tex]
[tex]d)~~\dfrac{x-1}{x-2} -\dfrac{x+1}{x^{2} -4} =\dfrac{x^{2}-3 }{x^{2} -4}~~,~~D=R-\{-2,2\}[/tex]
Szczegółowe wyjaśnienie:
Mamy wykonać działania na ułamkach i doprowadzić je do najprostszej postaci.
- pamiętamy o założeniach [tex]\boxed{\dfrac{x}{y} ,~~zal.~~y\neq 0}[/tex]
- korzystamy ze wzoru skróconego mnożenia [tex]\boxed{x^{2} -y^{2} =(x+y)\cdot (x-y)}[/tex]
- pamiętamy, gdy przed nawiasem jest znak - ( minus ) to usuwając nawias zmieniamy znak każdemu z wyrazów znajdującemu się w nawiasie na przeciwny.
[tex]a)~~zal.~~x\neq -3,~~x\neq -2,~~x\neq 0~~\Rightarrow~~D=R-\{-3,-2,0 \}\\\\\dfrac{x^{2} -6x}{x+3} \div \dfrac{x}{x-2} =\dfrac{x\cdot (x -6)}{x+3} \cdot \dfrac{x+2}{x} =\dfrac{(x-6)(x+2)}{x+3} =\dfrac{x^{2} +2x-6x-12}{x+3} =\dfrac{x^{2} -4x-12}{x+3}[/tex]
[tex]b)~~zal.~~x+2\neq 0,~~x^{2} +5x\neq 0~~\Rightarrow~~D=R-\{-5,-2,0\}\\\\\dfrac{x}{x^{2} +5x} \cdot \dfrac{x^{2} -25}{x+2} =\dfrac{x}{x\cdot (x +5)} \cdot \dfrac{(x+5)\cdot (x-5)}{x+2}=\dfrac{x-5}{x+2}[/tex]
[tex]c)~~zal.~~x-1\neq 0,~~x+2\neq 0~~\Rightarrow~~D=R-\{-2,1\}\\\\\dfrac{x-2}{x-1} +\dfrac{x-3}{x+2} =\dfrac{(x-2)\cdot (x+2) +(x-3)\cdot (x-1)}{(x-1)\cdot (x+2)} =\dfrac{x^{2}-4+x^{2} -x-3x+3 }{ (x-1)\cdot (x+2) } =\dfrac{ 2x^{2} -4x-1}{ (x-1)\cdot (x+2) } =\dfrac{ 2x^{2} -4x-1}{ x^{2} +2x-x-2} =\dfrac{ 2x^{2} -4x-1}{ x^{2} +x-2}[/tex]
[tex]d)~~zal.~~x-2\neq 0,~~x^{2} -4\neq 0~~\Rightarrow~~D=R-\{-2,2\}\\\\\dfrac{x-1}{x-2} -\dfrac{x+1}{x^{2} -4} =\dfrac{x-1}{x-2} -\dfrac{x+1}{(x-2)\cdot (x+2)}=\dfrac{(x-1)\cdot (x+2)-(x+1)}{ (x-2)\cdot (x+2) } =\dfrac{x^{2} +2x-x-2-x-1}{ (x-2)\cdot (x+2) } =\dfrac{x^{2} -3}{ (x-2)\cdot (x+2) } =\dfrac{x^{2}-3 }{x^{2} -4}[/tex]
