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CAPITULO XIV

Operaciones con Fracciones

Ejercicio 127

Simplificar:

$\begin{matrix} \frac{1}{a + 1} + \frac{1}{a - 1} & = \frac{a - 1 + a + 1}{(a + 1) (a - 1)} \\ = \frac{2 a}{a^{2} - 1} \end{matrix}$
$\begin{matrix} \frac{2}{x + 4} + \frac{1}{x - 3} & = \frac{2 (x - 3) + x + 4}{(x + 4) (x - 3)} \\ = \frac{2 x - 6 + x + 4}{(x + 4) (x - 3)} \\ = \frac{3 x - 2}{(x + 4) (x - 3)} \end{matrix}$
$\begin{matrix} \frac{3}{1 - x} + \frac{6}{2 x + 5} & = \frac{3 (2 x + 5) + 6 (1 - x)}{(1 - x) (2 x + 5)} \\ = \frac{6 x + 15 + 6 - 6 x}{(1 - x) (2 x + 5)} \\ = \frac{21}{(1 - x) (2 x + 5)} \end{matrix}$
$\begin{matrix} \frac{x}{x - y} + \frac{x}{x + y} & = \frac{x (x + y) + x (x - y)}{(x - y) (x + y)} \\ = \frac{x [x + y + x - y]}{x^{2} - y^{2}} \\ = \frac{2 x^{2}}{x^{2} - y^{2}} \end{matrix}$
$\begin{matrix} \frac{m + 3}{m - 3} + \frac{m + 2}{m - 2} & = \frac{(m - 2) (m + 3) + (m + 2) (m - 3)}{(m - 3) (m - 2)} \\ = \frac{m^{2} + m - 6 + m^{2} - m - 6}{(m - 3) (m - 2)} \\ = \frac{2 m^{2} - 12}{(m - 3) (m - 2)} \\ = \frac{2 (m^{2} - 6)}{(m - 3) (m - 2)} \end{matrix}$
$\begin{matrix} \frac{x + y}{x - y} + \frac{x - y}{x + y} & = \frac{{(x + y)}^{2} + {(x - y)}^{2}}{(x - y) (x + y)} \\ = \frac{x^{2} + 2 x y + y^{2} + x^{2} - 2 x y + y^{2}}{x^{2} - y^{2}} \\ = \frac{2 x^{2} + 2 y^{2}}{x^{2} - y^{2}} \\ = \frac{2 (x^{2} + y^{2})}{x^{2} - y^{2}} \end{matrix}$
$\begin{matrix} \frac{x}{x^{2} - 1} + \frac{x + 1}{{(x - 1)}^{2}} & = \frac{x}{(x - 1) (x + 1)} + \frac{x + 1}{{(x - 1)}^{2}} \\ = \frac{x (x - 1) + {(x + 1)}^{2}}{(x + 1) {(x - 1)}^{2}} \\ = \frac{x^{2} - x + x^{2} + 2 x + 1}{(x + 1) {(x - 1)}^{2}} \\ = \frac{2 x^{2} + x + 1}{(x + 1) {(x - 1)}^{2}} \end{matrix}$
$\begin{matrix} \frac{2}{x - 5} + \frac{3 x}{x^{2} - 25} & = \frac{2}{x - 5} + \frac{3 x}{(x - 5) (x + 5)} \\ = \frac{2 (x + 5) + 3 x}{(x - 5) (x + 5)} \\ = \frac{2 x + 10 + 3 x}{(x - 5) (x + 5)} \\ = \frac{5 x + 10}{(x - 5) (x + 5)} \\ = \frac{5 (x + 2)}{(x - 5) (x + 5)} \end{matrix}$
$\begin{matrix} \frac{1}{3 x - 2 y} + \frac{x - y}{9 x^{2} - 4 y^{2}} & = \frac{1}{3 x - 2 y} + \frac{x - y}{(3 x - 2 y) (3 x + 2 y)} \\ = \frac{3 x + 2 y + x - y}{(3 x - 2 y) (3 x + 2 y)} \\ = \frac{4 x + y}{(3 x - 2 y) (3 x + 2 y)} \end{matrix}$
$\begin{matrix} \frac{x + a}{x + 3 a} + \frac{3 a^{2} - x^{2}}{x^{2} - 9 a^{2}} & = \frac{x + a}{x + 3 a} + \frac{3 a^{2} - x^{2}}{(x + 3 a) (x - 3 a)} \\ = \frac{(x + a) (x - 3 a) + 3 a^{2} - x^{2}}{(x + 3 a) (x - 3 a)} \\ = \frac{x^{2} - 2 a x - 3 a^{2} + 3 a^{2} - x^{2}}{(x + 3 a) (x - 3 a)} \\ = \frac{- 2 a x}{x^{2} - 9 a^{2}} \\ = \frac{2 a x}{9 a^{2} - x^{2}} \end{matrix}$
$\begin{matrix} \frac{a}{1 - a^{2}} + \frac{a}{1 + a^{2}} & = a [\frac{1 + a^{2} + 1 - a^{2}}{(1 - a^{2}) (1 + a^{2})}] \\ = a [\frac{2}{1 - a^{4}}] \\ = \frac{2 a}{1 - a^{4}} \end{matrix}$
$\begin{matrix} \frac{2}{a^{2} - a b} + \frac{2}{a b + b^{2}} & = 2 [\frac{1}{a (a - b)} + \frac{1}{b (a + b)}] \\ = 2 [\frac{b (a + b) + a (a - b)}{a b (a + b) (a - b)}] \\ = 2 [\frac{a b + b^{2} + a^{2} - a b}{a b (a^{2} - b^{2})}] \\ = \frac{2 (a^{2} + b^{2})}{a b (a^{2} - b^{2})} \end{matrix}$
$\begin{matrix} \frac{a b}{9 a^{2} - b^{2}} + \frac{a}{3 a + b} & = a [\frac{b}{(3 a + b) (3 a - b)} + \frac{1}{3 a + b}] \\ = a [\frac{b + 3 a - b}{(3 a + b) (3 a - b)}] \\ = \frac{3 a^{2}}{9 a^{2} - b^{2}} \end{matrix}$
$\begin{matrix} \frac{1}{a^{2} - b^{2}} + \frac{1}{{(a - b)}^{2}} & = \frac{1}{(a + b) (a - b)} + \frac{1}{{(a - b)}^{2}} \\ = \frac{a - b + a + b}{(a + b) {(a - b)}^{2}} \\ = \frac{2 a}{(a + b) {(a - b)}^{2}} \end{matrix}$
$\begin{matrix} \frac{3}{x^{2} + y^{2}} + \frac{2}{{(x + y)}^{2}} & = \frac{3 {(x + y)}^{2} + 2 (x^{2} + y^{2})}{(x^{2} + y^{2}) {(x + y)}^{2}} \\ = \frac{3 (x^{2} + 2 x y + y^{2}) + 2 x^{2} + 2 y^{2}}{(x^{2} + y^{2}) {(x + y)}^{2}} \\ = \frac{3 x^{2} + 6 x y + 3 y^{2} + 2 x^{2} + 2 y^{2}}{(x^{2} + y^{2}) {(x + y)}^{2}} \\ = \frac{5 x^{2} + 6 x y + 5 y^{2}}{(x^{2} + y^{2}) {(x + y)}^{2}} \end{matrix}$
$\begin{matrix} \frac{x}{a^{2} - a x} + \frac{a + x}{a x} + \frac{a}{a x - x^{2}} & = \frac{x}{a (a - x)} + \frac{a + x}{a x} + \frac{a}{x (a - x)} \\ = \frac{x^{2} + (a + x) (a - x) + a^{2}}{a x (a - x)} \\ = \frac{x^{2} + a^{2} - x^{2} + a^{2}}{a x (a - x)} \\ = \frac{2 a^{2}}{a x (a - x)} \\ = \frac{2 a}{x (a - x)} \end{matrix}$
$\begin{matrix} \frac{3}{2 x + 4} + \frac{x - 1}{2 x - 4} + \frac{x + 8}{x^{2} - 4} & = \frac{3}{2 (x + 2)} + \frac{x - 1}{2 (x - 2)} + \frac{x + 8}{(x + 2) (x - 2)} \\ = \frac{3 (x - 2) + (x - 1) (x + 2) + 2 (x + 8)}{2 (x + 2) (x - 2)} \\ = \frac{3 x - 6 + x^{2} + x - 2 + 2 x + 16}{2 (x + 2) (x - 2)} \\ = \frac{x^{2} + 6 x + 8}{2 (x + 2) (x - 2)} \\ = \frac{(x + 4) (x + 2)}{2 (x + 2) (x - 2)} \\ = \frac{x + 4}{2 (x - 2)} \end{matrix}$
$\begin{matrix} \frac{1}{x + x^{2}} + \frac{1}{x - x^{2}} + \frac{x + 3}{1 - x^{2}} & = \frac{1}{x (1 + x)} + \frac{1}{x (1 - x)} + \frac{x + 3}{(1 - x) (1 + x)} \\ = \frac{1 - x + 1 + x + x (x + 3)}{x (1 - x) (1 + x)} \\ = \frac{2 + x^{2} + 3 x}{x (1 - x) (1 + x)} \\ = \frac{x^{2} + 3 x + 2}{x (1 - x) (1 + x)} \\ = \frac{(x + 2) (x + 1)}{x (1 - x) (x + 1)} \\ = \frac{x + 2}{x (1 - x)} \end{matrix}$
$\begin{matrix} \frac{x - y}{x + y} + \frac{x + y}{x - y} + \frac{4 x y}{x^{2} - y^{2}} & = \frac{x - y}{x + y} + \frac{x + y}{x - y} + \frac{4 x y}{(x + y) (x - y)} \\ = \frac{{(x - y)}^{2} + {(x + y)}^{2} + 4 x y}{(x + y) (x - y)} \\ = \frac{x^{2} - 2 x y + y^{2} + x^{2} + 2 x y + y^{2} + 4 x y}{(x + y) (x - y)} \\ = \frac{2 x^{2} + 4 x y + 2 y^{2}}{(x + y) (x - y)} \\ = \frac{2 (x^{2} + 2 x y + y^{2})}{(x + y) (x - y)} \\ = \frac{2 {(x + y)}^{2}}{(x + y) (x - y)} \\ = \frac{2 (x + y)}{x - y} \end{matrix}$
$\begin{matrix} \frac{1}{a - 5} + \frac{a}{a^{2} - 4 a - 5} + \frac{a + 5}{a^{2} + 2 a + 1} & = \frac{1}{a - 5} + \frac{a}{(a - 5) (a + 1)} + \frac{a + 5}{{(a + 1)}^{2}} \\ = \frac{{(a + 1)}^{2} + a (a + 1) + (a + 5) (a - 5)}{(a - 5) {(a + 1)}^{2}} \\ = \frac{a^{2} + 2 a + 1 + a^{2} + a + a^{2} - 25}{(a - 5) {(a + 1)}^{2}} \\ = \frac{3 a^{2} + 3 a - 24}{(a - 5) {(a + 1)}^{2}} \\ = \frac{3 (a^{2} + a - 8)}{(a - 5) {(a + 1)}^{2}} \end{matrix}$
$\begin{matrix} \frac{3}{a} + \frac{2}{5 a - 3} + \frac{1 - 85 a}{25 a^{2} - 9} & = \frac{3}{a} + \frac{2}{5 a - 3} + \frac{1 - 85 a}{(5 a - 3) (5 a + 3)} \\ = \frac{3 (5 a - 3) (5 a + 3) + 2 a (5 a + 3) + a (1 - 85 a)}{a (5 a - 3) (5 a + 3)} \\ = \frac{3 (25 a^{2} - 9) + 10 a^{2} + 6 a + a - 85 a^{2}}{a (5 a - 3) (5 a + 3)} \\ = \frac{75 a^{2} - 27 + 7 a - 75 a^{2}}{a (25 a^{2} - 9)} \\ = \frac{7 a - 27}{a (25 a^{2} - 9)} \end{matrix}$
$\begin{matrix} \frac{x + 1}{10} + \frac{x - 3}{5 x - 10} + \frac{x - 2}{2} & = \frac{x + 1}{10} + \frac{x - 3}{5 (x - 2)} + \frac{x - 2}{2} \\ = \frac{(x + 1) (x - 2) + 2 (x - 3) + 5 {(x - 2)}^{2}}{10 (x - 2)} \\ = \frac{x^{2} - x - 2 + 2 x - 6 + 5 (x^{2} - 4 x + 4)}{10 (x - 2)} \\ = \frac{x^{2} + x - 8 + 5 x^{2} - 20 x + 20}{10 (x - 2)} \\ = \frac{6 x^{2} - 19 x + 12}{10 (x - 2)} \end{matrix}$
$\begin{matrix} \frac{x + 5}{x^{2} + x - 12} + \frac{x + 4}{x^{2} + 2 x - 15} + \frac{x - 3}{x^{2} + 9 x + 20} & = \frac{x + 5}{(x + 4) (x - 3)} + \frac{x + 4}{(x + 5) (x - 3)} + \frac{x - 3}{(x + 5) (x + 4)} \\ = \frac{{(x + 5)}^{2} + {(x + 4)}^{2} + {(x - 3)}^{2}}{(x + 4) (x - 3) (x + 5)} \\ = \frac{x^{2} + 10 x + 25 + x^{2} + 8 x + 16 + x^{2} - 6 x + 9}{(x + 4) (x - 3) (x + 5)} \\ = \frac{3 x^{2} + 12 x + 50}{(x + 4) (x - 3) (x + 5)} \end{matrix}$
$\begin{matrix} \frac{1}{x - 2} + \frac{1 - 2 x^{2}}{x^{3} - 8} + \frac{x}{x^{2} + 2 x + 4} & = \frac{1}{x - 2} + \frac{1 - 2 x^{2}}{(x - 2) (x^{2} + 2 x + 4)} + \frac{x}{x^{2} + 2 x + 4} \\ = \frac{x^{2} + 2 x + 4 + 1 - 2 x^{2} + x (x - 2)}{(x - 2) (x^{2} + 2 x + 4)} \\ = \frac{x^{2} + 2 x + 4 + 1 - 2 x^{2} + x^{2} - 2 x}{x^{3} - 8} \\ = \frac{5}{x^{3} - 8} \end{matrix}$
$\begin{matrix} \frac{2}{a + 1} + \frac{a}{{(a + 1)}^{2}} + \frac{a + 1}{{(a + 1)}^{3}} & = \frac{1}{a + 1} [2 + \frac{a}{a + 1} + \frac{a + 1}{{(a + 1)}^{2}}] \\ = \frac{1}{a + 1} [\frac{2 {(a + 1)}^{2} + a (a + 1) + a + 1}{{(a + 1)}^{2}}] \\ = \frac{1}{a + 1} [\frac{2 (a^{2} + 2 a + 1) + a^{2} + a + a + 1}{{(a + 1)}^{2}}] \\ = \frac{1}{a + 1} [\frac{2 a^{2} + 4 a + 2 + a^{2} + 2 a + 1}{{(a + 1)}^{2}}] \\ = \frac{1}{a + 1} [\frac{3 a^{2} + 6 a + 3}{{(a + 1)}^{2}}] \\ = \frac{3}{a + 1} [\frac{a^{2} + 2 a + 1}{{(a + 1)}^{2}}] \\ = \frac{3}{a + 1} [\frac{{(a + 1)}^{2}}{{(a + 1)}^{2}}] \\ = \frac{3}{a + 1} \end{matrix}$
$\begin{matrix} \frac{2 x}{3 x^{2} + 11 x + 6} + \frac{x + 1}{x^{2} - 9} + \frac{1}{3 x + 2} & = \frac{2 x}{3 x^{2} + 9 x + 2 x + 6} + \frac{x + 1}{(x - 3) (x + 3)} + \frac{1}{3 x + 2} \\ = \frac{2 x}{3 x (x + 3) + 2 (x + 3)} + \frac{x + 1}{(x - 3) (x + 3)} + \frac{1}{3 x + 2} \\ = \frac{2 x}{(3 x + 2) (x + 3)} + \frac{x + 1}{(x - 3) (x + 3)} + \frac{1}{3 x + 2} \\ = \frac{2 x (x - 3) + (x + 1) (3 x + 2) + (x + 3) (x - 3)}{(3 x + 2) (x + 3) (x - 3)} \\ = \frac{6 x^{2} - x - 7}{(3 x + 2) (x + 3) (x - 3)} \end{matrix}$
$\begin{matrix} \frac{x^{2} - 4}{x^{3} + 1} + \frac{1}{x + 1} + \frac{3}{x^{2} - x + 1} & = \frac{x^{2} - 4}{(x + 1) (x^{2} - x + 1)} + \frac{1}{x + 1} + \frac{3}{x^{2} - x + 1} \\ = \frac{x^{2} - 4 + x^{2} - x + 1 + 3 (x + 1)}{(x + 1) (x^{2} - x + 1)} \\ = \frac{2 x^{2} - 4 - x + 1 + 3 x + 3}{(x + 1) (x^{2} - x + 1)} \\ = \frac{2 x^{2} + 2 x}{(x + 1) (x^{2} - x + 1)} \\ = \frac{2 x (x + 1)}{(x + 1) (x^{2} - x + 1)} \\ = \frac{2 x}{x^{2} - x + 1} \end{matrix}$
$\begin{matrix} \frac{1}{x - 1} + \frac{1}{(x - 1) (x + 2)} + \frac{x + 1}{(x - 1) (x + 2) (x + 3)} & = \frac{1}{x - 1} [1 + \frac{1}{x + 2} + \frac{x + 1}{(x + 2) (x + 3)}] \\ = \frac{1}{x - 1} [\frac{(x + 2) (x + 3) + x + 3 + x + 1}{(x + 2) (x + 3)}] \\ = \frac{1}{x - 1} [\frac{x^{2} + 5 x + 6 + 2 x + 4}{(x + 2) (x + 3)}] \\ = \frac{1}{x - 1} [\frac{x^{2} + 7 x + 10}{(x + 2) (x + 3)}] \\ = \frac{1}{x - 1} [\frac{(x + 2) (x + 5)}{(x + 2) (x + 3)}] \\ = \frac{x + 5}{(x - 1) (x + 3)} \end{matrix}$
$\begin{matrix} \frac{x - 2}{2 x^{2} - 5 x - 3} + \frac{x - 3}{2 x^{2} - 3 x - 2} + \frac{2 x - 1}{x^{2} - 5 x + 6} & = \frac{x - 2}{2 x^{2} + x - 6 x - 3} + \frac{x - 3}{2 x^{2} - 4 x + x - 2} + \frac{2 x - 1}{(x - 3) (x - 2)} \\ = \frac{x - 2}{x (2 x + 1) - 3 (2 x + 1)} + \frac{x - 3}{2 x (x - 2) + (x - 2)} + \frac{2 x - 1}{(x - 3) (x - 2)} \\ = \frac{x - 2}{(x - 3) (2 x + 1)} + \frac{x - 3}{(2 x + 1) (x - 2)} + \frac{2 x - 1}{(x - 3) (x - 2)} \\ = \frac{{(x - 2)}^{2} + {(x - 3)}^{2} + (2 x + 1) (2 x - 1)}{(x - 3) (2 x + 1) (x - 2)} \\ = \frac{x^{2} - 4 x + 4 + x^{2} - 6 x + 9 + 4 x^{2} - 1}{(x - 3) (2 x + 1) (x - 2)} \\ = \frac{6 x^{2} - 10 x + 12}{(x - 3) (2 x + 1) (x - 2)} \\ = \frac{2 (3 x^{2} - 5 x + 6)}{(x - 3) (2 x + 1) (x - 2)} \end{matrix}$
$\begin{matrix} \frac{a - 2}{a - 1} + \frac{a + 3}{a + 2} + \frac{a + 1}{a - 3} & = \frac{(a - 2) (a + 2) (a - 3) + (a - 1) (a - 3) (a + 3) + (a - 1) (a + 1) (a + 2)}{(a - 1) (a + 2) (a - 3)} \\ = \frac{(a^{2} - 4) (a - 3) + (a - 1) (a^{2} - 9) + (a^{2} - 1) (a + 2)}{(a - 1) (a + 2) (a - 3)} \\ = \frac{a^{3} - 4 a - 3 a^{2} + 12 + a^{3} - 9 a - a^{2} + 9 + a^{3} + 2 a^{2} - a - 2}{(a - 1) (a + 2) (a - 3)} \\ = \frac{3 a^{3} - 2 a^{2} - 14 a + 19}{(a - 1) (a + 2) (a - 3)} \end{matrix}$