$\frac{1}{a} + \frac{1}{a + b} = \frac{a + b + a}{a(a + b)} = \frac{2a + b}{a(a + b)}$

$\frac{1}{a + b} + \frac{1}{a - b} = \frac{a - b + a + b}{(a - b)(a + b)} = \frac{2a}{a^2 - b^2}$

$\frac{1}{m + n} - \frac{1}{n} = \frac{n - (m + n)}{n(m + n)} = \frac{n - m - n}{n(m + n)} = \frac{-m}{n(m + n)} = -\frac{m}{n(m + n)}$

$\frac{1}{x - y} - \frac{1}{x + y} = \frac{x + y - (x - y)}{(x - y)(x + y)} = \frac{x + y - x + y}{x^2 - y^2} = \frac{2y}{x^2 - y^2}$

$\frac{2}{a - b} + \frac{3}{a + b} = \frac{2(a + b) + 3(a - b)}{a^2 - b^2} = \frac{2a + 2b + 3a - 3b}{(a - b)(a + b)} = \frac{5a - b}{a^2 - b^2}$

$\frac{4}{p - q} - \frac{3}{p + q} = \frac{4(p + q) - 3(p - q)}{p^2 - q^2} = \frac{4p + 4q - 3p + 3q}{p^2 - q^2} = \frac{p + 7q}{p^2 - q^2}$

$\frac{2a}{a - 2b} + \frac{3a}{a + b} = \frac{2a(a + b) + 3a(a - 2b)}{(a - 2b)(a + b)} = \frac{2a^2 + 2ab + 3a^2 - 6ab}{(a - 2b)(a + b)} = \frac{5a^2 - 4ab}{(a - 2b)(a + b)}$

$\frac{3x}{x - y} - \frac{2x}{2x - y} = \frac{3x(2x - y) - 2x(x - y)}{(x - y)(2x - y)} = \frac{6x^2 - 3xy - 2x^2 + 2xy)}{(x - y)(2x - y)} = \frac{4x^2 - xy}{(x - y)(2x - y)}$

$\frac{5m}{2m - n} - \frac{3m}{n - m} = \frac{5m(n - m) - 3m(2m - n)}{(2m - n)(n - m)} = \frac{5mn - 5m^2 - 6m^2 + 3mn}{(2m - n)(n - m)} = \frac{-11m^2 + 8mn}{(2m - n)(n - m)}$

$\frac{4p}{q - 2p} - \frac{2p}{2p + q} = \frac{4p(2p + q) - 2p(q - 2p)}{(q - 2p)(q + 2p)} = \frac{8p^2 + 4pq - 2pq + 4p^2}{q^2 - 4p^2} = \frac{12p^2 + 2pq}{q^2 - 4p^2}$

$\frac{7}{2x - y} - \frac{5}{y - 2x} = \frac{7}{2x - y} + \frac{5}{2x - y} = \frac{7 + 5}{2x - y} = \frac{12}{2x - y}$

$\frac{5x}{x - 3y} + \frac{4x + 3y}{3y - x} = \frac{5x}{x - 3y} - \frac{4x + 3y}{x - 3y} = \frac{5x - (4x + 3y)}{x - 3y} = \frac{5x - 4x - 3y}{x - 3y} = \frac{x - 3y}{x - 3y} = 1$