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CAPITULO XIV

Operaciones con Fracciones

Ejercicio 138

Simplificar:

$\begin{matrix} \frac{1 + \frac{x + 1}{x - 1}}{\frac{1}{x - 1} - \frac{1}{x + 1}} & = \frac{\frac{x - 1 + x + 1}{x - 1}}{\frac{x + 1 - x + 1}{(x + 1) (x - 1)}} \\ = \frac{2 x (x + 1)}{2} \\ = x (x + 1) \end{matrix}$
$\begin{matrix} \frac{\frac{1}{x - 1} + \frac{2}{x + 1}}{\frac{x - 2}{x} + \frac{2 x + 6}{x + 1}} & = \frac{\frac{x + 1 + 2 (x - 1)}{(x + 1) (x - 1)}}{\frac{(x - 2) (x + 1) + x (2 x + 6)}{x (x + 1)}} \\ = \frac{\frac{x + 1 + 2 x - 2}{x - 1}}{\frac{x^{2} - x - 2 + 2 x^{2} + 6 x}{x}} \\ = \frac{\frac{3 x - 1}{x - 1}}{\frac{3 x^{2} + 5 x - 2}{x}} \\ = \frac{\frac{3 x - 1}{x - 1}}{\frac{3 x^{2} + 6 x - x - 2}{x}} \\ = \frac{\frac{3 x - 1}{x - 1}}{\frac{3 x (x + 2) - (x + 2)}{x}} \\ = \frac{\frac{3 x - 1}{x - 1}}{\frac{(3 x - 1) (x + 2)}{x}} \\ = \frac{x}{(x - 1) (x + 2)} \end{matrix}$
$\begin{matrix} \frac{\frac{a}{a - b} - \frac{b}{a + b}}{\frac{a + b}{a - b} + \frac{a}{b}} & = \frac{\frac{a (a + b) - b (a - b)}{(a - b) (a + b)}}{\frac{b (a + b) + a (a - b)}{b (a - b)}} \\ = \frac{\frac{a^{2} + a b - a b + b^{2}}{a + b}}{\frac{a b + b^{2} + a^{2} - a b}{b}} \\ = \frac{b (a^{2} + b^{2})}{(a^{2} + b^{2}) (a + b)} \\ = \frac{b}{a + b} \end{matrix}$
$\begin{matrix} \frac{\frac{x + 3}{x + 4} - \frac{x + 1}{x + 2}}{\frac{x - 1}{x + 2} - \frac{x - 3}{x + 4}} & = \frac{\frac{(x + 3) (x + 2) - (x + 4) (x + 1)}{(x + 4) (x + 2)}}{\frac{(x - 1) (x + 4) - (x - 3) (x + 2)}{(x + 4) (x + 2)}} \\ = \frac{x^{2} + 5 x + 6 - (x^{2} + 5 x + 4)}{x^{2} + 3 x - 4 - (x^{2} - x - 6)} \\ = \frac{x^{2} + 5 x + 6 - x^{2} - 5 x - 4}{x^{2} + 3 x - 4 - x^{2} + x + 6} \\ = \frac{2}{4 x + 2} \\ = \frac{2}{2 (2 x + 1)} \\ = \frac{1}{2 x + 1} \end{matrix}$
$\begin{matrix} \frac{\frac{m^{2}}{n} - \frac{m^{2} - n^{2}}{m + n}}{\frac{m - n}{n} + \frac{n}{m}} & = \frac{\frac{m^{2} (m + n) - n (m^{2} - n^{2})}{n (m + n)}}{\frac{m (m - n) + n^{2}}{n m}} \\ = \frac{\frac{m^{3} + m^{2} n - m^{2} n + n^{3}}{m + n}}{\frac{m^{2} - m n + n^{2}}{m}} \\ = \frac{m (m^{3} + n^{3})}{(m + n) (m^{2} - m n + n^{2})} \\ = m \end{matrix}$
$\begin{matrix} \frac{\frac{a^{2}}{b^{3}} + \frac{1}{a}}{\frac{a}{b} - \frac{b - a}{a - b}} & = \frac{\frac{a^{3} + b^{3}}{a b^{}}}{\frac{a (a - b) - b (b - a)}{b (a - b)}} \\ = \frac{\frac{a^{3} + b^{3}}{a b^{2}}}{\frac{a^{2} - a b - b^{2} + a b}{a - b}} \\ = \frac{(a^{3} + b^{3}) (a - b)}{a b^{2} (a^{2} - b^{2})} \\ = \frac{(a + b) (a^{2} - a b + b^{2}) (a - b)}{a b^{2} (a^{2} - b^{2})} \\ = \frac{a^{2} - a b + b^{2}}{a b^{2}} \end{matrix}$
$\begin{matrix} \frac{1 + \frac{2 x}{1 + x^{2}}}{2 x + \frac{2 x^{5} + 2}{1 - x^{4}}} & = \frac{\frac{1 + x^{2} + 2 x}{1 + x^{2}}}{\frac{2 x (1 - x^{4}) + 2 x^{5} + 2}{1 - x^{4}}} \\ = \frac{\frac{{(x + 1)}^{2}}{1 + x^{2}}}{\frac{2 x - 2 x^{5} + 2 x^{5} + 2}{(1 + x^{2}) (1 - x^{2})}} \\ = \frac{{(x + 1)}^{2} (1 - x^{2})}{2 x + 2} \\ = \frac{{(x + 1)}^{2} (1 - x^{2})}{2 (x + 1)} \\ = \frac{(x + 1) (1 - x^{2})}{2} \end{matrix}$
$\begin{matrix} \frac{\frac{x + y}{x - y} - \frac{x - y}{x + y}}{\frac{x + y}{x} - \frac{x + 2 y}{x + y}} & = \frac{\frac{{(x + y)}^{2} - {(x - y)}^{2}}{(x - y) (x + y)}}{\frac{{(x + y)}^{2} - x (x + 2 y)}{x (x + y)}} \\ = \frac{\frac{[(x + y) - (x - y)] [(x + y) + (x - y)]}{x - y}}{\frac{x^{2} + 2 x y + y^{2} - x^{2} - 2 x y}{x}} \\ = \frac{\frac{(x + y - x + y) (x + y + x - y)}{x - y}}{\frac{y^{2}}{x}} \\ = \frac{\frac{(2 y) (2 x)}{x - y}}{\frac{y^{2}}{x}} \\ = \frac{4 x^{2}}{y (x - y)} \end{matrix}$
$\begin{matrix} \frac{\frac{a + x}{a - x} - \frac{b + x}{b - x}}{\frac{2}{a - x} - \frac{2}{b - x}} & = \frac{\frac{(a + x) (b - x) - (a - x) (b + x)}{(a - x) (b - x)}}{\frac{2 (b - x) - 2 (a - x)}{(a - x) (b - x)}} \\ = \frac{a b - a x + b x - x^{2} - (a b + a x - b x - x^{2})}{2 (b - x - a + x)} \\ = \frac{a b - a x + b x - x^{2} - a b - a x + b x + x^{2}}{2 (b - a)} \\ = \frac{2 b x - 2 a x}{2 (b - a)} \\ = \frac{2 x (b - a)}{2 (b - a)} \\ = x \end{matrix}$
$\begin{matrix} \frac{\frac{a}{a + x} - \frac{a}{2 a + 2 x}}{\frac{a}{a - x} + \frac{a}{a + x}} & = \frac{\frac{a}{a + x} [1 - \frac{1}{2}]}{a [\frac{1}{a - x} + \frac{1}{a + x}]} \\ = \frac{\frac{a}{a + x} [\frac{2 - 1}{2}]}{a [\frac{a + x + a - x}{(a + x) (a - x)}]} \\ = \frac{\frac{a}{2 (a + x)}}{\frac{2 a^{2}}{(a + x) (a - x)}} \\ = \frac{a - x}{4 a} \end{matrix}$
$\begin{matrix} \frac{\frac{a + 2 b}{a - b} + \frac{b}{a}}{\frac{a + b}{a} + \frac{3 b}{a - b}} & = \frac{\frac{a (a + 2 b) + b (a - b)}{a (a - b)}}{\frac{(a + b) (a - b) + 3 a b}{a (a - b)}} \\ = \frac{a^{2} + 2 a b + a b - b^{2}}{a^{2} - b^{2} + 3 a b} \\ = \frac{a^{2} + 3 a b - b^{2}}{a^{2} + 3 a b - b^{2}} \\ = 1 \end{matrix}$
$\begin{matrix} \frac{1 - \frac{7}{x} + \frac{12}{x^{2}}}{x - \frac{16}{x}} & = \frac{\frac{x^{2} - 7 x + 12}{x^{2}}}{\frac{x^{2} - 16}{x}} \\ = \frac{\frac{(x - 3) (x - 4)}{x}}{(x - 4) (x + 4)} \\ = \frac{x - 3}{x (x + 4)} \end{matrix}$
$\begin{matrix} \frac{\frac{a^{2}}{b} - \frac{b^{2}}{a}}{\frac{1}{b} + \frac{1}{a} + \frac{b}{a^{2}}} & = \frac{\frac{a^{3} - b^{3}}{a b}}{\frac{a^{2} + a b + b^{2}}{a^{2} b}} \\ = \frac{a (a^{3} - b^{3})}{a^{2} + a b + b^{2}} \\ = \frac{a (a - b) (a^{2} + a b + b^{2})}{a^{2} + a b + b^{2}} \\ = a (a - b) \end{matrix}$
$\begin{matrix} \frac{x - 2 y - \frac{4 y^{2}}{x + y}}{x - 3 y - \frac{5 y^{2}}{x + y}} & = \frac{\frac{(x - 2 y) (x + y) - 4 y^{2}}{x + y}}{\frac{(x - 3 y) (x + y) - 5 y^{2}}{x + y}} \\ = \frac{x^{2} - x y - 2 y^{2} - 4 y^{2}}{x^{2} - 2 x y - 3 y^{2} - 5 y^{2}} \\ = \frac{x^{2} - x y - 6 y^{2}}{x^{2} - 2 x y - 8 y^{2}} \\ = \frac{x^{2} + 2 x y - 3 x y - 6 y^{2}}{x^{2} - 4 x y + 2 x y - 8 y^{2}} \\ = \frac{x (x + 2 y) - 3 y (x + 2 y)}{x (x - 4 y) + 2 y (x - 4 y)} \\ = \frac{(x - 3 y) (x + 2 y)}{(x + 2 y) (x - 4 y)} \\ = \frac{x - 3 y}{x - 4 y} \end{matrix}$
$\begin{matrix} \frac{\frac{2}{1 - a} + \frac{2}{1 + a}}{\frac{2}{1 + a} - \frac{2}{1 - a}} & = \frac{2 [\frac{1}{1 - a} + \frac{1}{1 + a}]}{2 [\frac{1}{1 + a} - \frac{1}{1 - a}]} \\ = \frac{\frac{1 + a + 1 - a}{(1 - a) (1 + a)}}{\frac{1 - a - (1 + a)}{(1 - a) (1 + a)}} \\ = \frac{2}{1 - a - 1 - a} \\ = - \frac{2}{2 a} \\ = - \frac{1}{a} \end{matrix}$
$\begin{matrix} \frac{\frac{1}{x + y + z} - \frac{1}{x - y + z}}{\frac{1}{x - y + z} - \frac{1}{x + y + z}} & = \frac{\frac{x - y + z - (x + y + z)}{(x + y + z) (x - y + z)}}{\frac{x + y + z - (x - y + z)}{(x + y + z) (x - y + z)}} \\ = \frac{x - y + z - x - y - z}{x + y + z - x + y - z} \\ = - \frac{2 y}{2 y} \\ = - 1 \end{matrix}$
$\begin{matrix} \frac{1 + \frac{2 b + c}{a - b - c}}{1 - \frac{c - 2 b}{a - b + c}} & = \frac{\frac{a - b - c + 2 b + c}{a - b - c}}{\frac{a - b + c - (c - 2 b)}{a - b + c}} \\ = \frac{\frac{a + b}{a - b - c}}{\frac{a - b + c - c + 2 b}{a - b + c}} \\ = \frac{\frac{a + b}{a - b - c}}{\frac{a + b}{a - b + c}} \\ = \frac{a - b + c}{a - b - c} \end{matrix}$
$\begin{matrix} \frac{\frac{a}{1 - a} + \frac{1 - a}{a}}{\frac{1 - a}{a} - \frac{a}{1 - a}} & = \frac{\frac{a^{2} + {(1 - a)}^{2}}{a (1 - a)}}{\frac{{(1 - a)}^{2} - a^{2}}{a (1 - a)}} \\ = \frac{a^{2} + 1 - 2 a + a^{2}}{1 - 2 a + a^{2} - a^{2}} \\ = \frac{2 a^{2} - 2 a + 1}{1 - 2 a} \end{matrix}$
$\begin{matrix} \frac{\frac{x + 1 - \frac{6 x + 12}{x + 2}}{x - 5}}{\frac{x - 4 + \frac{11 x - 22}{x - 2}}{x + 7}} & = \frac{\frac{\frac{(x + 1) (x + 2) - (6 x + 12)}{x + 2}}{x - 5}}{\frac{\frac{(x - 4) (x - 2) + 11 x - 22}{x - 2}}{x + 7}} \\ = \frac{\frac{\frac{x^{2} - 3 x - 10}{x + 2}}{x - 5}}{\frac{\frac{x^{2} + 5 x - 14}{x - 2}}{x + 7}} \\ = \frac{\frac{\frac{(x - 5) (x + 2)}{x + 2}}{x - 5}}{\frac{\frac{(x + 7) (x - 2)}{x - 2}}{x + 7}} \\ = 1 \end{matrix}$
$\begin{matrix} \frac{1}{1 + \frac{1}{x}} & = \frac{1}{\frac{x + 1}{x}} \\ = \frac{x}{x + 1} \end{matrix}$
$\begin{matrix} \frac{1}{1 + \frac{1}{1 - \frac{1}{x}}} & = \frac{1}{1 + \frac{1}{\frac{x - 1}{x}}} \\ = \frac{1}{1 + \frac{x}{x - 1}} \\ = \frac{1}{\frac{x - 1 + x}{x - 1}} \\ = \frac{1}{\frac{2 x - 1}{x - 1}} \\ = \frac{x - 1}{2 x - 1} \end{matrix}$
$\begin{matrix} 1 - \frac{1}{2 + \frac{1}{\frac{x}{3} - 1}} & = 1 - \frac{1}{2 + \frac{1}{\frac{x - 3}{3}}} \\ = 1 - \frac{1}{2 + \frac{3}{x - 3}} \\ = 1 - \frac{1}{\frac{2 x - 6 + 3}{x - 3}} \\ = 1 - \frac{x - 3}{2 x - 3} \\ = \frac{2 x - 3 - (x - 3)}{2 x - 3} \\ = \frac{2 x - 3 - x + 3}{2 x - 3} \\ = \frac{x}{2 x - 3} \end{matrix}$
$\begin{matrix} \frac{2}{1 + \frac{2}{1 + \frac{2}{x}}} & = \frac{2}{1 + \frac{2}{\frac{x + 2}{x}}} \\ = \frac{2}{1 + \frac{2 x}{x + 2}} \\ = \frac{2}{\frac{x + 2 + 2 x}{x + 2}} \\ = \frac{2}{\frac{3 x + 2}{x + 2}} \\ = \frac{2 (x + 2)}{3 x + 2} \end{matrix}$
$\begin{matrix} \frac{1}{x - \frac{x}{x - \frac{x^{2}}{x + 1}}} & = \frac{1}{x - \frac{x}{\frac{x (x + 1) - x^{2}}{x + 1}}} \\ = \frac{1}{x - \frac{x}{\frac{x^{2} + x - x^{2}}{x + 1}}} \\ = \frac{1}{x - \frac{x (x + 1)}{x}} \\ = \frac{1}{x - x - 1} \\ = - 1 \end{matrix}$
$\begin{matrix} \frac{1}{a + 2 - \frac{a + 1}{a - \frac{1}{a}}} & = \frac{1}{a + 2 - \frac{a + 1}{\frac{a^{2} - 1}{a}}} \\ = \frac{1}{a + 2 - \frac{a (a + 1)}{a^{2} - 1}} \\ = \frac{1}{a + 2 - \frac{a (a + 1)}{(a + 1) (a - 1)}} \\ = \frac{1}{a + 2 - \frac{a}{a - 1}} \\ = \frac{1}{\frac{(a + 2) (a - 1) - a}{a - 1}} \\ = \frac{1}{\frac{a^{2} + a - 2 - a}{a - 1}} \\ = \frac{a - 1}{a^{2} - 2} \end{matrix}$
$\begin{matrix} \frac{x - 1}{x + 2 - \frac{x^{2} + 2}{x - \frac{x - 2}{x + 1}}} & = \frac{x - 1}{x + 2 - \frac{x^{2} + 2}{\frac{x (x + 1) - (x - 2)}{x + 1}}} \\ = \frac{x - 1}{x + 2 - \frac{x^{2} + 2}{\frac{x^{2} + x - x + 2}{x + 1}}} \\ = \frac{x - 1}{x + 2 - \frac{x^{2} + 2}{\frac{x^{2} + 2}{x + 1}}} \\ = \frac{x - 1}{x + 2 - (x + 1)} \\ = \frac{x - 1}{x + 2 - x - 1} \\ = x - 1 \end{matrix}$