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CAPITULO XIV

Operaciones con Fracciones

Ejercicio 130

Simplificar:

$\begin{matrix} \frac{2}{x - 3} + \frac{3}{x + 2} - \frac{4 x - 7}{x^{2} - x - 6} & = \frac{2}{x - 3} + \frac{3}{x + 2} - \frac{4 x - 7}{(x - 3) (x + 2)} \\ = \frac{2 (x + 2) + 3 (x - 3) - (4 x - 7)}{(x - 3) (x + 2)} \\ = \frac{2 x + 4 + 3 x - 9 - 4 x + 7}{(x - 3) (x + 2)} \\ = \frac{x + 2}{(x - 3) (x + 2)} \\ = \frac{1}{x - 3} \end{matrix}$
$\begin{matrix} \frac{a}{3 a + 6} - \frac{1}{6 a + 12} + \frac{a + 12}{12 a + 24} & = \frac{a}{3 (a + 2)} - \frac{1}{6 (a + 2)} + \frac{a + 12}{12 (a + 2)} \\ = \frac{1}{3 (a + 2)} [a - \frac{1}{2} + \frac{a + 12}{4}] \\ = \frac{1}{3 (a + 2)} [\frac{4 a - 2 + a + 12}{4}] \\ = \frac{1}{3 (a + 2)} [\frac{5 a + 10}{4}] \\ = \frac{1}{3 (a + 2)} [\frac{5 (a + 2)}{4}] \\ = \frac{5}{12} \end{matrix}$
$\begin{matrix} \frac{x}{x^{2} + 1} + \frac{1}{3 x} - \frac{1}{x^{2}} & = \frac{3 x^{3} + x (x^{2} + 1) - 3 (x^{2} + 1)}{3 x^{2} (x^{2} + 1)} \\ = \frac{3 x^{3} + x^{3} + x - 3 x^{2} - 3}{3 x^{2} (x^{2} + 1)} \\ = \frac{4 x^{3} - 3 x^{2} + x - 3}{3 x^{2} (x^{2} + 1)} \end{matrix}$
$\begin{matrix} \frac{a + 3}{a^{2} - 1} + \frac{a - 1}{2 a + 2} + \frac{a - 4}{4 a - 4} & = \frac{a + 3}{(a - 1) (a + 1)} + \frac{a - 1}{2 (a + 1)} + \frac{a - 4}{4 (a - 1)} \\ = \frac{4 (a + 3) + 2 {(a - 1)}^{2} + (a + 1) (a - 4)}{4 (a - 1) (a + 1)} \\ = \frac{4 a + 12 + 2 (a^{2} - 2 a + 1) + a^{2} - 3 a - 4}{4 (a - 1) (a + 1)} \\ = \frac{a + 8 + 2 a^{2} - 4 a + 2 + a^{2}}{4 (a^{2} - 1)} \\ = \frac{3 a^{2} - 3 a + 10}{4 (a^{2} - 1)} \end{matrix}$
$\begin{matrix} \frac{a - b}{a^{2} + a b} + \frac{a + b}{a b} - \frac{a}{a b + b^{2}} & = \frac{a - b}{a (a + b)} + \frac{a + b}{a b} - \frac{a}{b (a + b)} \\ = \frac{b (a - b) + {(a + b)}^{2} - a^{2}}{a b (a + b)} \\ = \frac{a b - b^{2} + a^{2} + 2 a b + b^{2} - a^{2}}{a b (a + b)} \\ = \frac{3 a b}{a b (a + b)} \\ = \frac{3}{a + b} \end{matrix}$
$\begin{matrix} \frac{x - y}{x + y} - \frac{x + y}{x - y} + \frac{4 x^{2}}{x^{2} - y^{2}} & = \frac{x - y}{x + y} - \frac{x + y}{x - y} + \frac{4 x^{2}}{(x + y) (x - y)} \\ = \frac{{(x - y)}^{2} - {(x + y)}^{2} + 4 x^{2}}{(x + y) (x - y)} \\ = \frac{[(x - y) + (x + y)] [(x - y) - (x + y)] + 4 x^{2}}{(x + y) (x - y)} \\ = \frac{[x - y + x + y] [x - y - x - y] + 4 x^{2}}{(x + y) (x - y)} \\ = \frac{2 x (- 2 y) + 4 x^{2}}{(x + y) (x - y)} \\ = \frac{4 x^{2} - 4 x y}{(x + y) (x - y)} \\ = \frac{4 x (x - y)}{(x + y) (x - y)} \\ = \frac{4 x}{x + y} \end{matrix}$
$\begin{matrix} \frac{x}{a^{2} - a x} + \frac{1}{a} + \frac{1}{x} & = \frac{x}{a (a - x)} + \frac{1}{a} + \frac{1}{x} \\ = \frac{x^{2} + x (a - x) + a (a - x)}{a x (a - x)} \\ = \frac{x^{2} + a x - x^{2} + a^{2} - a x}{a x (a - x)} \\ = \frac{a^{2}}{a x (a - x)} \\ = \frac{a}{x (a - x)} \end{matrix}$
$\begin{matrix} \frac{x + 1}{x^{2} - x - 20} - \frac{x + 4}{x^{2} - 4 x - 5} + \frac{x + 5}{x^{2} + 5 x + 4} & = \frac{x + 1}{(x - 5) (x + 4)} - \frac{x + 4}{(x - 5) (x + 1)} + \frac{x + 5}{(x + 4) (x + 1)} \\ = \frac{{(x + 1)}^{2} - {(x + 4)}^{2} + (x - 5) (x + 5)}{(x + 1) (x - 5) (x + 4)} \\ = \frac{[(x + 1) + (x + 4)] [(x + 1) - (x + 4)] + x^{2} - 25}{(x + 1) (x - 5) (x + 4)} \\ = \frac{[x + 1 + x + 4] [x + 1 - x - 4] + x^{2} - 25}{(x + 1) (x - 5) (x + 4)} \\ = \frac{- 3 (2 x + 5) + x^{2} - 25}{(x + 1) (x - 5) (x + 4)} \\ = \frac{- 6 x - 15 + x^{2} - 25}{(x + 1) (x - 5) (x + 4)} \\ = \frac{x^{2} - 6 x - 40}{(x + 1) (x - 5) (x + 4)} \\ = \frac{(x - 10) (x + 4)}{(x + 1) (x - 5) (x + 4)} \\ = \frac{x - 10}{(x + 1) (x - 5)} \end{matrix}$
$\begin{matrix} \frac{2 x + 1}{12 x + 8} - \frac{x^{2}}{6 x^{2} + x - 2} + \frac{2 x}{16 x - 8} & = \frac{2 x + 1}{4 (3 x + 2)} - \frac{x^{2}}{6 x^{2} - 3 x + 4 x - 2} + \frac{2 x}{8 (2 x - 1)} \\ = \frac{2 x + 1}{4 (3 x + 2)} - \frac{x^{2}}{3 x (2 x - 1) + 2 (2 x - 1)} + \frac{2 x}{8 (2 x - 1)} \\ = \frac{2 x + 1}{4 (3 x + 2)} - \frac{x^{2}}{(3 x + 2) (2 x - 1)} + \frac{2 x}{8 (2 x - 1)} \\ = \frac{2 (2 x - 1) (2 x + 1) - 8 x^{2} + 2 x (3 x + 2)}{8 (3 x + 2) (2 x - 1)} \\ = \frac{2 (4 x^{2} - 1) - 8 x^{2} + 6 x^{2} + 4 x}{8 (3 x + 2) (2 x - 1)} \\ = \frac{8 x^{2} - 2 - 2 x^{2} + 4 x}{8 (3 x + 2) (2 x - 1)} \\ = \frac{6 x^{2} + 4 x - 2}{8 (3 x + 2) (2 x - 1)} \\ = \frac{2 (3 x^{2} + 2 x - 1)}{(3 x + 2) (2 x - 1)} \\ = \frac{3 x^{2} + 2 x - 1}{4 (3 x + 2) (2 x - 1)} \end{matrix}$
$\begin{matrix} \frac{1}{a x} - \frac{1}{a^{2} + a x} + \frac{1}{a + x} & = \frac{1}{a x} - \frac{1}{a (a + x)} + \frac{1}{a + x} \\ = \frac{a + x - x + a x}{a x (a + x)} \\ = \frac{a + a x}{a x (a + x)} \\ = \frac{a (1 + x)}{a x (a + x)} \\ = \frac{1 + x}{x (a + x)} \end{matrix}$
$\begin{matrix} \frac{1}{x + y} - \frac{1}{x - y} + \frac{2 y}{x^{2} + y^{2}} & = \frac{(x - y) (x^{2} + y^{2}) - (x + y) (x^{2} + y^{2}) + 2 y (x + y) (x - y)}{(x + y) (x - y) (x^{2} + y^{2})} \\ = \frac{x^{3} + x y^{2} - x^{2} y - y^{3} - (x^{3} + x y^{2} + x^{2} y + y^{3}) + 2 y (x^{2} - y^{2})}{(x + y) (x - y) (x^{2} + y^{2})} \\ = \frac{x^{3} + x y^{2} - x^{2} y - y^{3} - x^{3} - x y^{2} - x^{2} y - y^{3} + 2 x^{2} y - 2 y^{3}}{(x^{2} - y^{2}) (x^{2} + y^{2})} \\ = \frac{- 4 y^{3}}{x^{4} - y^{4}} \\ = \frac{4 y^{3}}{y^{4} - x^{4}} \end{matrix}$
$\begin{matrix} \frac{a - 1}{3 a + 3} - \frac{a - 2}{6 a - 6} + \frac{a^{2} + 2 a - 6}{9 a^{2} - 9} & = \frac{a - 1}{3 (a + 1)} - \frac{a - 2}{6 (a - 1)} + \frac{a^{2} + 2 a - 6}{9 (a^{2} - 1)} \\ = \frac{1}{3} [\frac{a - 1}{a + 1} - \frac{a - 2}{2 (a - 1)} + \frac{a^{2} + 2 a - 6}{3 (a - 1) (a + 1)}] \\ = \frac{1}{3} [\frac{6 {(a - 1)}^{2} - 3 (a - 2) (a + 1) + 2 (a^{2} + 2 a - 6)}{6 (a - 1) (a + 1)}] \\ = \frac{1}{3} [\frac{6 (a^{2} - 2 a + 1) - 3 (a^{2} - a - 2) + 2 a^{2} + 4 a - 12}{6 (a - 1) (a + 1)}] \\ = \frac{1}{3} [\frac{6 a^{2} - 12 a + 6 - 3 a^{2} + 3 a + 6 + 2 a^{2} + 4 a - 12}{6 (a - 1) (a + 1)}] \\ = \frac{1}{3} [\frac{5 a^{2} - 5 a}{6 (a - 1) (a + 1)}] \\ = \frac{1}{3} [\frac{5 a (a - 1)}{6 (a - 1) (a + 1)}] \\ = \frac{5 a}{18 (a + 1)} \end{matrix}$
$\begin{matrix} \frac{1}{a^{2} + 2 a - 24} + \frac{2}{a^{2} - 2 a - 8} - \frac{3}{a^{2} + 8 a + 12} & = \frac{1}{(a + 6) (a - 4)} + \frac{2}{(a - 4) (a + 2)} - \frac{3}{(a + 6) (a + 2)} \\ = \frac{a + 2 + 2 (a + 6) - 3 (a - 4)}{(a + 6) (a - 4) (a + 2)} \\ = \frac{a + 2 + 2 a + 12 - 3 a + 12}{(a + 6) (a - 4) (a + 2)} \\ = \frac{26}{(a + 6) (a - 4) (a + 2)} \end{matrix}$
$\begin{matrix} \frac{x + y}{x y} - \frac{x + 2 y}{x y + y^{2}} - \frac{y}{x^{2} + x y} & = \frac{x + y}{x y} - \frac{x + 2 y}{y (x + y)} - \frac{y}{x (x + y)} \\ = \frac{{(x + y)}^{2} - x (x + 2 y) - y^{2}}{x y (x + y)} \\ = \frac{x^{2} + 2 x y + y^{2} - x^{2} - 2 x y - y^{2}}{x y (x + y)} \\ = 0 \end{matrix}$
$\begin{matrix} \frac{a^{3}}{a^{3} + 1} + \frac{a + 3}{a^{2} - a + 1} - \frac{a - 1}{a + 1} & = \frac{a^{3}}{(a + 1) (a^{2} - a + 1)} + \frac{a + 3}{a^{2} - a + 1} - \frac{a - 1}{a + 1} \\ = \frac{a^{3} + (a + 3) (a + 1) - (a - 1) (a^{2} - a + 1)}{(a + 1) (a^{2} - a + 1)} \\ = \frac{a^{3} + a^{2} + 4 a + 3 - (a^{3} - a^{2} + a - a^{2} + a - 1)}{(a + 1) (a^{2} - a + 1)} \\ = \frac{a^{3} + a^{2} + 4 a + 3 - a^{3} + 2 a^{2} - 2 a + 1}{a^{3} + 1} \\ = \frac{3 a^{2} + 2 a + 4}{a^{3} + 1} \end{matrix}$
$\begin{matrix} \frac{1}{x - 1} + \frac{2 x}{x^{2} - 1} - \frac{3 x^{2}}{x^{3} - 1} & = \frac{1}{x - 1} + \frac{2 x}{(x - 1) (x + 1)} - \frac{3 x^{2}}{(x - 1) (x^{2} + x + 1)} \\ = \frac{1}{x - 1} [1 + \frac{2 x}{x + 1} - \frac{3 x^{2}}{x^{2} + x + 1}] \\ = \frac{1}{x - 1} [\frac{(x + 1) (x^{2} + x + 1) + 2 x (x^{2} + x + 1) - 3 x^{2} (x + 1)}{(x + 1) (x^{2} + x + 1)}] \\ = \frac{1}{x - 1} [\frac{x^{3} + x^{2} + x + x^{2} + x + 1 + 2 x^{3} + 2 x^{2} + 2 x - 3 x^{3} - 3 x^{2}}{(x + 1) (x^{2} + x + 1)}] \\ = \frac{1}{x - 1} [\frac{x^{2} + 4 x + 1}{(x + 1) (x^{2} + x + 1)}] \\ = \frac{x^{2} + 4 x + 1}{(x + 1) (x^{3} - 1)} \end{matrix}$
$\begin{matrix} \frac{a + b}{a^{2} - a b + b^{2}} - \frac{1}{a + b} + \frac{3 a^{2}}{a^{3} + b^{3}} & = \frac{a + b}{a^{2} - a b + b^{2}} - \frac{1}{a + b} + \frac{3 a^{2}}{(a + b) (a^{2} - a b + b^{2})} \\ = \frac{{(a + b)}^{2} - (a^{2} - a b + b^{2}) + 3 a^{2}}{(a + b) (a^{2} - a b + b^{2})} \\ = \frac{a^{2} + 2 a b + b^{2} - a^{2} + a b - b^{2} + 3 a^{2}}{(a + b) (a^{2} - a b + b^{2})} \\ = \frac{3 a^{2} + 3 a b}{(a + b) (a^{2} - a b + b^{2})} \\ = \frac{3 a (a + b)}{(a + b) (a^{2} - a b + b^{2})} \\ = \frac{3 a}{a^{2} - a b + b^{2}} \end{matrix}$
$\begin{matrix} \frac{2}{x - 2} + \frac{2 x + 3}{x^{2} + 2 x + 4} - \frac{6 x + 12}{x^{3} - 8} & = \frac{2}{x - 2} + \frac{2 x + 3}{x^{2} + 2 x + 4} - \frac{6 x + 12}{(x - 2) (x^{2} + 2 x + 4)} \\ = \frac{2 (x^{2} + 2 x + 4) + (x - 2) (2 x + 3) - (6 x + 12)}{(x - 2) (x^{2} + 2 x + 4)} \\ = \frac{2 x^{2} + 4 x + 8 + 2 x^{2} + 3 x - 4 x - 6 - 6 x - 12}{(x - 2) (x^{2} + 2 x + 4)} \\ = \frac{4 x^{2} - 3 x - 10}{(x - 2) (x^{2} + 2 x + 4)} \\ = \frac{4 x^{2} - 8 x + 5 x - 10}{(x - 2) (x^{2} + 2 x + 4)} \\ = \frac{4 x (x - 2) + 5 (x - 2)}{(x - 2) (x^{2} + 2 x + 4)} \\ = \frac{(4 x + 5) (x - 2)}{(x - 2) (x^{2} + 2 x + 4)} \\ = \frac{4 x + 5}{x^{2} + 2 x + 4} \end{matrix}$
$\begin{matrix} \frac{3 x + 2}{x^{2} + 3 x - 10} - \frac{5 x + 1}{x^{2} + 4 x - 5} + \frac{4 x - 1}{x^{2} - 3 x + 2} & = \frac{3 x + 2}{(x + 5) (x - 2)} - \frac{5 x + 1}{(x + 5) (x - 1)} + \frac{4 x - 1}{(x - 2) (x - 1)} \\ = \frac{(3 x + 2) (x - 1) - (x - 2) (5 x + 1) + (x + 5) (4 x - 1)}{(x + 5) (x - 2) (x - 1)} \\ = \frac{3 x^{2} + 2 x - 3 x + 2 - 5 x^{2} - x + 10 x + 2 + 4 x^{2} - x + 20 x - 5}{(x + 5) (x - 2) (x - 1)} \\ = \frac{2 x^{2} + 27 x - 1}{(x + 5) (x - 2) (x - 1)} \end{matrix}$
$\begin{matrix} \frac{1}{{(n - 1)}^{2}} + \frac{1}{n - 1} - \frac{1}{{(n - 1)}^{3}} - \frac{1}{n} & = \frac{1}{{(n - 1)}^{2}} + \frac{1}{n - 1} - \frac{1}{{(n - 1)}^{3}} - \frac{1}{n} \\ = \frac{n (n - 1) + n {(n - 1)}^{2} - n - {(n - 1)}^{3}}{n {(n - 1)}^{3}} \\ = \frac{n^{2} - n + n (n^{2} - 2 n + 1) - n - (n^{3} - 3 n^{2} + 3 n - 1)}{n {(n - 1)}^{3}} \\ = \frac{n^{2} - 2 n + n^{3} - 2 n^{2} + n - n^{3} + 3 n^{2} - 3 n + 1}{n {(n - 1)}^{3}} \\ = \frac{2 n^{2} - 4 n + 1}{n {(n - 1)}^{3}} \end{matrix}$
$\begin{matrix} \frac{1}{a^{2} + 5} - \frac{a^{2} - 5}{{(a^{2} + 5)}^{2}} + \frac{a^{2} + 5}{a^{4} - 25} & = \frac{1}{a^{2} + 5} - \frac{a^{2} - 5}{{(a^{2} + 5)}^{2}} + \frac{a^{2} + 5}{(a^{2} + 5) (a^{2} - 5)} \\ = \frac{1}{a^{2} + 5} [1 - \frac{a^{2} - 5}{a^{2} + 5} + \frac{a^{2} + 5}{a^{2} - 5}] \\ = \frac{1}{a^{2} + 5} [\frac{(a^{2} + 5) (a^{2} - 5) - {(a^{2} - 5)}^{2} + {(a^{2} + 5)}^{2}}{(a^{2} + 5) (a^{2} - 5)}] \\ = \frac{1}{a^{2} + 5} [\frac{a^{4} - 25 - (a^{4} - 10 a^{2} + 25) + a^{4} + 10 a^{2} + 25}{(a^{2} + 5) (a^{2} - 5)}] \\ = \frac{1}{a^{2} + 5} [\frac{2 a^{4} - a^{4} + 10 a^{2} - 25 + 10 a^{2}}{(a^{2} + 5) (a^{2} - 5)}] \\ = \frac{1}{a^{2} + 5} [\frac{a^{4} + 20 a^{2} - 25}{(a^{2} + 5) (a^{2} - 5)}] \\ = \frac{a^{4} + 20 a^{2} - 25}{(a^{2} + 5) (a^{4} - 25)} \end{matrix}$
$\begin{matrix} \frac{1 - x^{2}}{9 - x^{2}} - \frac{x^{2}}{9 + 6 x + x^{2}} - \frac{6 x}{9 - 6 x + x^{2}} & = \frac{1 - x^{2}}{(3 - x) (3 + x)} - \frac{x^{2}}{{(3 + x)}^{2}} - \frac{6 x}{{(3 - x)}^{2}} \\ = \frac{(1 - x^{2}) (3 - x) (3 + x) - x^{2} {(3 - x)}^{2} - 6 x {(3 + x)}^{2}}{{(3 - x)}^{2} {(3 + x)}^{2}} \\ = \frac{(1 - x^{2}) (9 - x^{2}) - x^{2} (9 - 6 x + x^{2}) - 6 x (9 + 6 x + x^{2})}{{(3 - x)}^{2} {(3 + x)}^{2}} \\ = \frac{9 - x^{2} - 9 x^{2} + x^{4} - 9 x^{2} + 6 x^{3} - x^{4} - 54 x - 36 x^{2} - 6 x^{3}}{{(3 - x)}^{2} {(3 + x)}^{2}} \\ = \frac{9 - 54 x - 55 x^{2}}{{(3 - x)}^{2} {(3 + x)}^{2}} \end{matrix}$
$\begin{matrix} \frac{x}{2 x + 2} - \frac{x + 1}{3 x - 3} + \frac{x - 1}{6 x + 6} - \frac{5}{18 x - 18} & = \frac{x}{2 (x + 1)} - \frac{x + 1}{3 (x - 1)} + \frac{x - 1}{6 (x + 1)} - \frac{5}{18 (x - 1)} \\ = \frac{9 x (x - 1) - 6 {(x + 1)}^{2} + 3 {(x - 1)}^{2} - 5 (x + 1)}{18 (x - 1) (x + 1)} \\ = \frac{9 x^{2} - 9 x - 6 (x^{2} + 2 x + 1) + 3 (x^{2} - 2 x + 1) - 5 x - 5}{18 (x - 1) (x + 1)} \\ = \frac{9 x^{2} - 9 x - 6 x^{2} - 12 x - 6 + 3 x^{2} - 6 x + 3 - 5 x - 5}{18 (x - 1) (x + 1)} \\ = \frac{6 x^{2} - 32 x - 8}{18 (x - 1) (x + 1)} \\ = \frac{2 (3 x^{2} - 16 x - 4)}{(x - 1) (x + 1)} \\ = \frac{3 x^{2} - 16 x - 4}{9 (x - 1) (x + 1)} \end{matrix}$
$\begin{matrix} \frac{a + 2}{2 a + 2} - \frac{7 a}{8 a^{2} - 8} - \frac{a - 3}{4 a - 4} & = \frac{a + 2}{2 (a + 1)} - \frac{7 a}{8 (a^{2} - 1)} - \frac{a - 3}{4 (a - 1)} \\ = \frac{1}{2} [\frac{a + 2}{a + 1} - \frac{7 a}{4 (a + 1) (a - 1)} - \frac{a - 3}{2 (a - 1)}] \\ = \frac{1}{2} [\frac{4 (a + 2) (a - 1) - 7 a - 2 (a - 3) (a + 1)}{4 (a + 1) (a - 1)}] \\ = \frac{1}{2} [\frac{4 (a^{2} + a - 2) - 7 a - 2 (a^{2} - 2 a - 3)}{4 (a + 1) (a - 1)}] \\ = \frac{1}{2} [\frac{4 a^{2} + 4 a - 8 - 7 a - 2 a^{2} + 4 a + 6}{4 (a + 1) (a - 1)}] \\ = \frac{1}{2} [\frac{2 a^{2} + a - 2}{4 (a + 1) (a - 1)}] \\ = \frac{2 a^{2} + a - 2}{8 (a + 1) (a - 1)} \end{matrix}$
$\begin{matrix} \frac{a - 3}{20 a + 10} + \frac{2 a + 5}{40 a + 20} - \frac{4 a - 1}{60 a + 30} & = \frac{a - 3}{10 (2 a + 1)} + \frac{2 a + 5}{20 (2 a + 1)} - \frac{4 a - 1}{30 (2 a + 1)} \\ = \frac{1}{10 (2 a + 1)} [a - 3 + \frac{2 a + 5}{2} - \frac{4 a - 1}{3}] \\ = \frac{1}{10 (2 a + 1)} [\frac{6 a - 18 + 3 (2 a + 5) - 2 (4 a - 1)}{6}] \\ = \frac{1}{10 (2 a + 1)} [\frac{6 a - 18 + 6 a + 15 - 8 a + 2}{6}] \\ = \frac{1}{10 (2 a + 1)} [\frac{4 a - 1}{6}] \\ = \frac{4 a - 1}{60 (2 a + 1)} \end{matrix}$
$\begin{matrix} \frac{2}{2 x^{2} + 5 x + 3} - \frac{1}{2 x^{2} - x - 6} + \frac{3}{x^{2} - x - 2} & = \frac{2}{2 x^{2} + 2 x + 3 x + 3} - \frac{1}{2 x^{2} - 4 x + 3 x - 6} + \frac{3}{(x - 2) (x + 1)} \\ = \frac{2}{2 x (x + 1) + 3 (x + 1)} - \frac{1}{2 x (x - 2) + 3 (x - 2)} + \frac{3}{(x - 2) (x + 1)} \\ = \frac{2}{(2 x + 3) (x + 1)} - \frac{1}{(2 x + 3) (x - 2)} + \frac{3}{(x - 2) (x + 1)} \\ = \frac{2 (x - 2) - (x + 1) + 3 (2 x + 3)}{(2 x + 3) (x + 1) (x - 2)} \\ = \frac{2 x - 4 - x - 1 + 6 x + 9}{(2 x + 3) (x + 1) (x - 2)} \\ = \frac{7 x + 4}{(2 x + 3) (x + 1) (x - 2)} \end{matrix}$
$\begin{matrix} \frac{a - 1}{a - 2} - \frac{a - 2}{a + 3} + \frac{1}{a - 1} & = \frac{(a + 3) {(a - 1)}^{2} - {(a - 2)}^{2} (a - 1) + (a - 2) (a + 3)}{(a - 2) (a + 3) (a - 1)} \\ = \frac{(a + 3) (a^{2} - 2 a + 1) - (a^{2} - 4 a + 4) (a - 1) + a^{2} + a - 6}{(a - 2) (a + 3) (a - 1)} \\ = \frac{a^{3} - 2 a^{2} + a + 3 a^{2} - 6 a + 3 - (a^{3} - 4 a^{2} + 4 a - a^{2} + 4 a - 4) + a^{2} + a - 6}{(a - 2) (a + 3) (a - 1)} \\ = \frac{a^{3} + a^{2} - 5 a + 3 - a^{3} + 5 a^{2} - 8 a + 4 + a^{2} + a - 6}{(a - 2) (a + 3) (a - 1)} \\ = \frac{7 a^{2} - 12 a + 1}{(a - 2) (a + 3) (a - 1)} \end{matrix}$
$\begin{matrix} \frac{2 + 3 a}{2 - 3 a} - \frac{2 - 3 a}{2 + 3 a} - \frac{a}{{(2 - 3 a)}^{2}} & = \frac{{(2 + 3 a)}^{2} (2 - 3 a) - {(2 - 3 a)}^{3} - a (2 + 3 a)}{(2 + 3 a) {(2 - 3 a)}^{2}} \\ = \frac{(2 + 3 a) (4 - 9 a^{2}) - (8 - 36 a + 54 a^{2} - 27 a^{3}) - 2 a - 3 a^{2}}{(2 + 3 a) {(2 - 3 a)}^{2}} \\ = \frac{8 - 18 a^{2} + 12 a - 27 a^{3} - 8 + 36 a - 54 a^{2} + 27 a^{3} - 2 a - 3 a^{2}}{(2 + 3 a) {(2 - 3 a)}^{2}} \\ = \frac{46 a - 75 a^{2}}{(2 + 3 a) {(2 - 3 a)}^{2}} \end{matrix}$
$\begin{matrix} \frac{1}{5 + 5 a} + \frac{1}{5 - 5 a} - \frac{1}{10 + 10 a^{2}} & = \frac{1}{5 (1 + a)} + \frac{1}{5 (1 - a)} - \frac{1}{10 (1 + a^{2})} \\ = \frac{1}{5} [\frac{1}{1 + a} + \frac{1}{1 - a} - \frac{1}{2 (1 + a^{2})}] \\ = \frac{1}{5} [\frac{2 (1 + a^{2}) (1 - a) + 2 (1 + a^{2}) (1 + a) - (1 - a) (1 + a)}{2 (1 + a^{2}) (1 - a) (1 + a)}] \\ = \frac{1}{5} [\frac{2 (1 + a^{2} - a - a^{3}) + 2 (1 + a^{2} + a + a^{3}) - (1 - a^{2})}{2 (1 + a^{2}) (1 - a) (1 + a)}] \\ = \frac{1}{5} [\frac{2 + 2 a^{2} - 2 a - 2 a^{3} + 2 + 2 a^{2} + 2 a + 2 a^{3} - 1 + a^{2}}{2 (1 + a^{2}) (1 - a^{2})}] \\ = \frac{1}{5} [\frac{5 a^{2} + 3}{2 (1 + a^{2}) (1 - a^{2})}] \\ = \frac{5 a^{2} + 3}{10 (1 - a^{4})} \end{matrix}$
$\begin{matrix} \frac{1}{3 - 3 x} - \frac{1}{3 + 3 x} + \frac{x}{6 + 6 x^{2}} - \frac{x}{2 - 2 x^{2}} & = \frac{1}{3 (1 - x)} - \frac{1}{3 (1 + x)} + \frac{x}{6 (1 + x^{2})} - \frac{x}{2 (1 - x^{2})} \\ = \frac{2 (1 + x^{2}) (1 + x) - 2 (1 + x^{2}) (1 - x) + x (1 - x) (1 + x) - 3 x (1 + x^{2})}{6 (1 + x^{2}) (1 - x) (1 + x)} \\ = \frac{2 (1 + x^{2} + x + x^{3}) - 2 (1 + x^{2} - x - x^{3}) + x (1 - x^{2}) - 3 x - 3 x^{3}}{6 (1 + x^{2}) (1 - x^{2})} \\ = \frac{2 + 2 x^{2} + 2 x + 2 x^{3} - 2 - 2 x^{2} + 2 x + 2 x^{3} + x - x^{3} - 3 x - 3 x^{3}}{6 (1 - x^{4})} \\ = \frac{2 x}{6 (1 - x^{4})} \\ = \frac{2 x}{(1 - x^{4})} \\ = \frac{x}{3 (1 - x^{4})} \end{matrix}$