Вариант 1 Упростите выражение: a) $$x - \frac{x}{x+1}$$ b) $$\frac{m+2}{4m} - \frac{1}{m+4}$$ v) $$\frac{x}{x+y} + \frac{y}{x-y}$$ g) $$\frac{3x+y}{x^2+xy} - \frac{x+3y}{y^2+xy}$$

a) $$x - \frac{x}{x+1} = \frac{x(x+1) - x}{x+1} = \frac{x^2 + x - x}{x+1} = \frac{x^2}{x+1}$$ b) $$\frac{m+2}{4m} - \frac{1}{m+4} = \frac{(m+2)(m+4) - 4m}{4m(m+4)} = \frac{m^2 + 6m + 8 - 4m}{4m(m+4)} = \frac{m^2 + 2m + 8}{4m(m+4)}$$ v) $$\frac{x}{x+y} + \frac{y}{x-y} = \frac{x(x-y) + y(x+y)}{(x+y)(x-y)} = \frac{x^2 - xy + yx + y^2}{(x+y)(x-y)} = \frac{x^2 + y^2}{x^2 - y^2}$$ g) $$\frac{3x+y}{x^2+xy} - \frac{x+3y}{y^2+xy} = \frac{3x+y}{x(x+y)} - \frac{x+3y}{y(x+y)} = \frac{y(3x+y) - x(x+3y)}{xy(x+y)} = \frac{3xy + y^2 - x^2 - 3xy}{xy(x+y)} = \frac{y^2 - x^2}{xy(x+y)} = \frac{-(x^2 - y^2)}{xy(x+y)} = \frac{-(x-y)(x+y)}{xy(x+y)} = \frac{-(x-y)}{xy} = \frac{y-x}{xy}$$