Упражнение 151

\({\largeа)}\ \left(\frac{x}{x+1}+1\right)\cdot\frac{1+x}{2x-1}=\frac{x+x+1}{x+1}\cdot\frac{1+x}{2x-1}=\frac{ (2x+1)\cdot(x+1)}{ (x+1)\cdot(2x-1)}=\frac{2x+1}{2x-1};\)

\({\largeб)}\ \left(\frac{5y^2}{1-y^2}\right):\left(1-\frac{1}{1-y}\right)=\frac{5y^2}{1-y^2}:\frac{1-y-1}{1-y}=\frac{5y^2}{1-y^2}:\frac{{-}y}{1-y}=\frac{5y^2}{1-y^2}\cdot\left({-}\frac{1-y}{y}\right)={-}\frac{5y^2\cdot(1-y)}{ (1-y)(1+y)\cdot{y}}={-}\frac{5y}{1+y};\)

\({\largeв)}\ \left(\frac{4a}{2-a}-a\right):\frac{a+2}{a-2}=\frac{4a-a(2-a)}{2-a}:\frac{a+2}{a-2}=\frac{4a-2a+a^2}{2-a}\cdot\frac{a-2}{a+2}=\frac{ (a^2+2a)\cdot(a-2)}{ (2-a)\cdot(a+2)}=\frac{ a(a+2)\cdot(a-2)}{ {-}(a-2)\cdot(a+2)}={-}a;\)

\({\largeг)}\ \frac{x-2}{x-3}\cdot\left(x+\frac{x}{2-x}\right)=\frac{x-2}{x-3}\cdot\frac{ x(2-x)+x}{2-x}=\frac{x-2}{x-3}\cdot\frac{2x-x^2+x}{2-x}=\frac{x-2}{x-3}\cdot\frac{{-}x^2+3x}{2-x}=\frac{x-2}{x-3}\cdot\frac{{-}x(x-3)}{ {-}(x-2)}=\frac{ (x-2)\cdot{x}(x-3)}{ (x-3)\cdot(x-2)}=x.\)