Ejercicio 131 – Solucionario Baldor

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CAPITULO XIV

Operaciones con Fracciones

Ejercicio 131

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$\begin{matrix} \frac{1}{m - n} + \frac{m}{n^{2} - m^{2}} & = \frac{1}{m - n} - \frac{m}{m^{2} - n^{2}} \\ = \frac{1}{m - n} - \frac{m}{(m - n) (m + n)} \\ = \frac{1}{m - n} [1 - \frac{m}{m + n}] \\ = \frac{1}{m - n} [\frac{m + n - m}{m + n}] \\ = \frac{n}{m^{2} - n^{2}} \end{matrix}$
$\begin{matrix} \frac{x^{2}}{x^{2} - x y} - \frac{2 x}{y - x} & = \frac{x^{2}}{x (x - y)} + \frac{2 x}{x - y} \\ = \frac{x}{x - y} + \frac{2 x}{x - y} \\ = \frac{x + 2 x}{x - y} \\ = \frac{3 x}{x - y} \end{matrix}$
$\begin{matrix} \frac{1}{2 x - x^{2}} + \frac{x}{x^{2} - 4} & = \frac{1}{x (2 - x)} - \frac{x}{4 - x^{2}} \\ = \frac{1}{x (2 - x)} - \frac{x}{(2 - x) (2 + x)} \\ = \frac{2 + x - x^{2}}{x (2 - x) (2 + x)} \\ = \frac{(2 - x) (1 + x)}{x (2 - x) (2 + x)} \\ = \frac{x + 1}{x (x + 2)} \end{matrix}$
$\begin{matrix} \frac{a + b}{a^{2} - a b} + \frac{a}{b^{2} - a^{2}} & = \frac{a + b}{a (a - b)} - \frac{a}{a^{2} - b^{2}} \\ = \frac{a + b}{a (a - b)} - \frac{a}{(a + b) (a - b)} \\ = \frac{1}{a - b} [\frac{a + b}{a} - \frac{a}{a + b}] \\ = \frac{1}{a - b} [\frac{{(a + b)}^{2} - a^{2}}{a (a + b)}] \\ = \frac{1}{a - b} [\frac{a^{2} + 2 a b + b^{2} - a^{2}}{a (a + b)}] \\ = \frac{2 a b + b^{2}}{a (a^{2} - b^{2})} \end{matrix}$
$\begin{matrix} \frac{x - 4}{x^{2} - 2 x - 3} - \frac{x}{6 - 2 x} & = \frac{x - 4}{(x - 3) (x + 1)} + \frac{x}{2 x - 6} \\ = \frac{x - 4}{(x - 3) (x + 1)} + \frac{x}{2 (x - 3)} \\ = \frac{1}{x - 3} [\frac{x - 4}{x + 1} + \frac{x}{2}] \\ = \frac{1}{x - 3} [\frac{2 (x - 4) + x (x + 1)}{2 (x + 1)}] \\ = \frac{1}{x - 3} [\frac{2 x - 8 + x^{2} + x}{2 (x + 1)}] \\ = \frac{1}{x - 3} [\frac{x^{2} + 3 x - 8}{2 (x + 1)}] \\ = \frac{x^{2} + 3 x - 8}{2 (x + 1) (x - 3)} \end{matrix}$
$\begin{matrix} \frac{1}{x^{2} + 2 x - 8} + \frac{1}{(2 - x) (x + 3)} & = \frac{1}{(x + 4) (x - 2)} - \frac{1}{(x - 2) (x + 3)} \\ = \frac{1}{x - 2} [\frac{1}{x + 4} - \frac{1}{x + 3}] \\ = \frac{1}{x - 2} [\frac{x + 3 - (x + 4)}{(x + 4) (x + 3)}] \\ = \frac{1}{x - 2} [\frac{x + 3 - x - 4}{(x + 4) (x + 3)}] \\ = - \frac{1}{(x - 2) (x + 4) (x + 3)} \end{matrix}$
$\begin{matrix} \frac{1}{2 x + 2} + \frac{2}{1 - x} + \frac{7}{4 x - 4} & = \frac{1}{2 (x + 1)} - \frac{2}{x - 1} + \frac{7}{4 (x - 1)} \\ = \frac{2 (x - 1) - 8 (x + 1) + 7 (x + 1)}{4 (x - 1) (x + 1)} \\ = \frac{2 x - 2 - 8 x - 8 + 7 x + 7}{4 (x^{2} - 1)} \\ = \frac{x - 3}{4 (x^{2} - 1)} \end{matrix}$
$\begin{matrix} \frac{2 a}{a + 3} + \frac{3 a}{a - 3} + \frac{2 a}{9 - a^{2}} & = \frac{2 a}{a + 3} + \frac{3 a}{a - 3} - \frac{2 a}{a^{2} - 9} \\ = a [\frac{2}{a + 3} + \frac{3}{a - 3} - \frac{2}{(a + 3) (a - 3)}] \\ = a [\frac{2 (a - 3) + 3 (a + 3) - 2}{(a + 3) (a - 3)}] \\ = a [\frac{2 a - 6 + 3 a + 9 - 2}{a^{2} - 9}] \\ = \frac{a (5 a + 1)}{a^{2} - 9} \end{matrix}$
$\begin{matrix} \frac{x + 3 y}{y + x} + \frac{3 y^{2}}{x^{2} - y^{2}} - \frac{x}{y - x} & = \frac{x + 3 y}{y + x} - \frac{3 y^{2}}{y^{2} - x^{2}} - \frac{x}{y - x} \\ = \frac{x + 3 y}{y + x} - \frac{3 y^{2}}{(y - x) (y + x)} - \frac{x}{y - x} \\ = \frac{(x + 3 y) (y - x) - 3 y^{2} - x (y + x)}{(y - x) (y + x)} \\ = \frac{x y - x^{2} + 3 y^{2} - 3 x y - 3 y^{2} - x y - x^{2}}{y^{2} - x^{2}} \\ = \frac{- 2 x^{2} - 3 x y}{y^{2} - x^{2}} \\ = \frac{2 x^{2} + 3 x y}{x^{2} - y^{2}} \end{matrix}$
$\begin{matrix} \frac{x}{x^{2} + 2 x - 3} + \frac{x - 3}{(1 - x) (x + 2)} + \frac{1}{x + 2} & = \frac{x}{(x + 3) (x - 1)} - \frac{x - 3}{(x - 1) (x + 2)} + \frac{1}{x + 2} \\ = \frac{x (x + 2) - (x + 3) (x - 3) + (x + 3) (x - 1)}{(x + 3) (x - 1) (x + 2)} \\ = \frac{x^{2} + 2 x - (x^{2} - 9) + x^{2} + 2 x - 3}{(x + 3) (x - 1) (x + 2)} \\ = \frac{2 x^{2} + 4 x - 3 - x^{2} + 9}{(x + 3) (x - 1) (x + 2)} \\ = \frac{x^{2} + 4 x + 6}{(x + 3) (x - 1) (x + 2)} \end{matrix}$
$\begin{matrix} \frac{3}{2 a + 2} - \frac{1}{4 a - 4} - \frac{4}{8 - 8 a^{2}} & = \frac{3}{2 (a + 1)} - \frac{1}{4 (a - 1)} + \frac{4}{8 a^{2} - 8} \\ = \frac{3}{2 (a + 1)} - \frac{1}{4 (a - 1)} + \frac{4}{8 (a^{2} - 1)} \\ = \frac{1}{2} [\frac{3}{a + 1} - \frac{1}{2 (a - 1)} + \frac{4}{4 (a + 1) (a - 1)}] \\ = \frac{1}{2} [\frac{6 (a - 1) - (a + 1) + 2}{2 (a + 1) (a - 1)}] \\ = \frac{1}{2} [\frac{6 a - 6 - a - 1 + 2}{2 (a + 1) (a - 1)}] \\ = \frac{1}{2} [\frac{5 a - 5}{2 (a + 1) (a - 1)}] \\ = \frac{1}{2} [\frac{5 (a - 1)}{2 (a + 1) (a - 1)}] \\ = \frac{5}{4 (a + 1)} \end{matrix}$
$\begin{matrix} \frac{1}{a - 3} + \frac{a + 1}{(3 - a) (a - 2)} + \frac{2}{(2 - a) (1 - a)} & = \frac{1}{a - 3} - \frac{a + 1}{(a - 3) (a - 2)} + \frac{2}{(a - 2) (a - 1)} \\ = \frac{(a - 2) (a - 1) - (a + 1) (a - 1) + 2 (a - 3)}{(a - 3) (a - 2) (a - 1)} \\ = \frac{a^{2} - 3 a + 2 - (a^{2} - 1) + 2 a - 6}{(a - 3) (a - 2) (a - 1)} \\ = \frac{a^{2} - 3 a + 2 - a^{2} + 1 + 2 a - 6}{(a - 3) (a - 2) (a - 1)} \\ = \frac{- a - 3}{(a - 3) (a - 2) (a - 1)} \\ = \frac{a + 3}{(3 - a) (a - 2) (a - 1)} \end{matrix}$
$\begin{matrix} \frac{2 x}{x - 1} + \frac{2 x^{3} + 2 x^{2}}{1 - x^{3}} + \frac{1}{x^{2} + x + 1} & = \frac{2 x}{x - 1} - \frac{2 x^{2} (x + 1)}{x^{3} - 1} + \frac{1}{x^{2} + x + 1} \\ = \frac{2 x}{x - 1} - \frac{2 x^{2} (x + 1)}{(x - 1) (x^{2} + x + 1)} + \frac{1}{x^{2} + x + 1} \\ = \frac{2 x (x^{2} + x + 1) - 2 x^{2} (x + 1) + x - 1}{(x - 1) (x^{2} + x + 1)} \\ = \frac{2 x [(x^{2} + x + 1) - x (x + 1)] + x - 1}{(x - 1) (x^{2} + x + 1)} \\ = \frac{2 x [x^{2} + x + 1 - x^{2} - x] + x - 1}{(x - 1) (x^{2} + x + 1)} \\ = \frac{3 x - 1}{x^{3} - 1} \end{matrix}$
$\begin{matrix} \frac{x + 2}{3 x - 1} + \frac{x + 1}{3 - 2 x} + \frac{4 x^{2} + 6 x + 3}{6 x^{2} - 11 x + 3} & = \frac{x + 2}{3 x - 1} - \frac{x + 1}{2 x - 3} + \frac{4 x^{2} + 6 x + 3}{6 x^{2} - 2 x - 9 x + 3} \\ = \frac{x + 2}{3 x - 1} - \frac{x + 1}{2 x - 3} + \frac{4 x^{2} + 6 x + 3}{2 x (3 x - 1) - 3 (3 x - 1)} \\ = \frac{x + 2}{3 x - 1} - \frac{x + 1}{2 x - 3} + \frac{4 x^{2} + 6 x + 3}{(2 x - 3) (3 x - 1)} \\ = \frac{(2 x - 3) (x + 2) - (x + 1) (3 x - 1) + 4 x^{2} + 6 x + 3}{(2 x - 3) (3 x - 1)} \\ = \frac{(2 x^{2} - 3 x + 4 x - 6) - (3 x^{2} - x + 3 x - 1) + 4 x^{2} + 6 x + 3}{(2 x - 3) (3 x - 1)} \\ = \frac{2 x^{2} - 3 x + 4 x - 6 - 3 x^{2} + x - 3 x + 1 + 4 x^{2} + 6 x + 3}{(2 x - 3) (3 x - 1)} \\ = \frac{3 x^{2} + 5 x - 2}{(2 x - 3) (3 x - 1)} \\ = \frac{3 x^{2} - x + 6 x - 2}{(2 x - 3) (3 x - 1)} \\ = \frac{x (3 x - 1) + 2 (3 x - 1)}{(2 x - 3) (3 x - 1)} \\ = \frac{(x + 2) (3 x - 1)}{(2 x - 3) (3 x - 1)} \\ = \frac{x + 2}{2 x - 3} \end{matrix}$