Выполните действия в пункте а) из задания 149

149. Выполните действия: а) $$ \left(\frac{x}{x+1} + \frac{1+x}{2x-1}\right) \cdot \frac{1}{x} = \left(\frac{x(2x-1) + (1+x)(x+1)}{(x+1)(2x-1)}\right) \cdot \frac{1}{x} = \frac{2x^2 - x + 1 + 2x + x^2}{(x+1)(2x-1)} \cdot \frac{1}{x} = \frac{3x^2 + x + 1}{(x+1)(2x-1)x} $$ б) $$ \frac{5y^2}{1-y^2} : \left(1 - \frac{1}{1-y}\right) = \frac{5y^2}{(1-y)(1+y)} : \left(\frac{1-y-1}{1-y}\right) = \frac{5y^2}{(1-y)(1+y)} : \frac{-y}{1-y} = \frac{5y^2}{(1-y)(1+y)} \cdot \frac{1-y}{-y} = \frac{5y^2}{-(1+y)y} = -\frac{5y}{1+y} $$ в) $$ \left(\frac{4a}{2-a} - a\right) : \frac{a+2}{a-2} = \left(\frac{4a - a(2-a)}{2-a}\right) : \frac{a+2}{a-2} = \frac{4a - 2a + a^2}{2-a} : \frac{a+2}{a-2} = \frac{a^2 + 2a}{2-a} : \frac{a+2}{a-2} = \frac{a(a+2)}{-(a-2)} \cdot \frac{a-2}{a+2} = -a $$ г) $$ \frac{x-y}{x} \cdot \frac{5y}{x^2 - xy} = \frac{x-y}{x} \cdot \frac{5y}{x(x-y)} = \frac{5y}{x^2} $$ д) $$ \frac{x-2}{x-3} \cdot \frac{x+2}{2-x} = \frac{x-2}{x-3} \cdot \frac{x+2}{-(x-2)} = -\frac{x+2}{x-3} $$ е) $$ \left(\frac{x+3}{x^2+9} \cdot \frac{x-3}{x+3}\right) + \frac{x-3}{x+3} = \frac{x-3}{x^2+9} + \frac{x-3}{x+3} = (x-3)\left(\frac{1}{x^2+9} + \frac{1}{x+3}\right) = (x-3)\left(\frac{x+3+x^2+9}{(x^2+9)(x+3)}\right) = \frac{(x-3)(x^2+x+12)}{(x^2+9)(x+3)} $$ 150. Упростите выражение: а) $$ \left(\frac{2m+1}{2m-1} - \frac{2m-1}{2m+1}\right) : \frac{4m}{10m-5} = \left(\frac{(2m+1)^2 - (2m-1)^2}{(2m-1)(2m+1)}\right) : \frac{4m}{5(2m-1)} = \left(\frac{4m^2+4m+1 - (4m^2-4m+1)}{(2m-1)(2m+1)}\right) : \frac{4m}{5(2m-1)} = \frac{4m^2+4m+1 - 4m^2+4m-1}{(2m-1)(2m+1)} : \frac{4m}{5(2m-1)} = \frac{8m}{(2m-1)(2m+1)} \cdot \frac{5(2m-1)}{4m} = \frac{8m \cdot 5}{4m(2m+1)} = \frac{2 \cdot 5}{2m+1} = \frac{10}{2m+1} $$ 151. Выполните действия: а) $$ \frac{a^2-9}{2a^2+1} \cdot \left(\frac{6a+1}{a-3} + \frac{6a-1}{a+3}\right) = \frac{(a-3)(a+3)}{2a^2+1} \cdot \left(\frac{(6a+1)(a+3) + (6a-1)(a-3)}{(a-3)(a+3)}\right) = \frac{(a-3)(a+3)}{2a^2+1} \cdot \left(\frac{6a^2+18a+a+3 + 6a^2-18a-a+3}{(a-3)(a+3)}\right) = \frac{1}{2a^2+1} \cdot (12a^2+6) = \frac{6(2a^2+1)}{2a^2+1} = 6 $$ б) $$ \left(\frac{5x+y}{x-5y} + \frac{5x-y}{x+5y}\right) : \frac{x^2+y^2}{x^2-25y^2} = \left(\frac{(5x+y)(x+5y) + (5x-y)(x-5y)}{(x-5y)(x+5y)}\right) : \frac{x^2+y^2}{x^2-25y^2} = \left(\frac{5x^2+25xy+xy+5y^2 + 5x^2-25xy-xy+5y^2}{(x-5y)(x+5y)}\right) : \frac{x^2+y^2}{x^2-25y^2} = \frac{10x^2+10y^2}{x^2-25y^2} : \frac{x^2+y^2}{x^2-25y^2} = \frac{10(x^2+y^2)}{x^2-25y^2} \cdot \frac{x^2-25y^2}{x^2+y^2} = 10 $$