Ejercicio 142 – Solucionario Baldor

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DESCARGA

CAPITULO XV

Ecuaciones fraccionarias de primer grado

Resolución de ecuaciones fraccionarias con denominadores compuestos

Ejercicio 142

Resolver las siguientes ecuaciones

$\begin{matrix} \frac{3}{5} + \frac{3}{2 x - 1} & = 0 \\ \frac{3 (2 x - 1) + 3 (5)}{5 (2 x - 1)} & = 0 \\ 6 x - 3 + 15 & = 0 \\ 6 x + 12 & = 0 \\ 6 x & = - 12 \\ x & = - \frac{}{6} \\ x & = - 2 \end{matrix}$
$\begin{matrix} \frac{2}{4 x - 1} & = \frac{3}{4 x + 1} \\ 2 (4 x + 1) & = 3 (4 x - 1) \\ 8 x + 2 & = 12 x - 3 \\ 8 x - 12 x & = - 3 - 2 \\ - 4 x & = - 5 \\ x & = \frac{5}{4} \end{matrix}$
$\begin{matrix} \frac{5}{x^{2} - 1} & = \frac{1}{x - 1} \\ \frac{5}{(x - 1) (x + 1)} & = \frac{1}{x - 1} \\ \frac{5}{x + 1} & = 1 \\ 5 & = x + 1 \\ 5 - 1 & = x \\ x & = 4 \end{matrix}$
$\begin{matrix} \frac{3}{x + 1} - \frac{1}{x^{2} - 1} & = 0 \\ \frac{3}{x + 1} & = \frac{1}{x^{2} - 1} \\ \frac{3}{x + 1} & = \frac{1}{(x + 1) (x - 1)} \\ 3 & = \frac{1}{x - 1} \\ 3 (x - 1) & = 1 \\ 3 x - 3 & = 1 \\ 3 x & = 1 + 3 \\ 3 x & = 4 \\ x & = \frac{4}{3} \end{matrix}$
$\begin{matrix} \frac{5 x + 8}{3 x + 4} & = \frac{5 x + 2}{3 x - 4} \\ (5 x + 8) (3 x - 4) & = (5 x + 2) (3 x + 4) \\ 15 x^{2} - 20 x + 24 x - 32 & = 15 x^{2} + 20 x + 6 x + 8 \\ 4 x & = 26 x + 8 + 32 \\ 4 x - 26 x & = 40 \\ - 22 x & = 40 \\ x & = - \frac{}{} \\ x & = - \frac{20}{11} \end{matrix}$
$\begin{matrix} \frac{10 x^{2} - 5 x + 8}{5 x^{2} + 9 x - 19} & = 2 \\ 10 x^{2} - 5 x + 8 & = 2 (5 x^{2} + 9 x - 19) \\ 10 x^{2} - 5 x + 8 & = 10 x^{2} + 18 x - 38 \\ - 5 x - 18 x & = - 38 - 8 \\ - 23 x & = - 46 \\ x & = \frac{-}{- 23} \\ x & = 2 \end{matrix}$
$\begin{matrix} \frac{1}{3 x - 3} + \frac{1}{4 x + 4} & = \frac{1}{12 x - 12} \\ \frac{1}{3 (x - 1)} + \frac{1}{4 (x + 1)} & = \frac{1}{12 (x - 1)} \\ \frac{4 (x + 1) + 3 (x - 1)}{12 (x - 1) (x + 1)} & = \frac{1}{12 (x - 1)} \\ \frac{4 (x + 1) + 3 (x - 1)}{x + 1} & = 1 \\ 4 x + 4 + 3 x - 3 & = x + 1 \\ 7 x + 1 & = x + 1 \\ 7 x - x & = 0 \\ 6 x & = 0 \\ x & = 0 \end{matrix}$
$\begin{matrix} \frac{x}{4} - \frac{x^{2} - 8 x}{4 x - 5} & = \frac{7}{4} \\ \frac{x (4 x - 5) - 4 (x^{2} - 8 x)}{4 (4 x - 5)} & = \frac{7}{4} \\ 4 x^{2} - 5 x - 4 x^{2} + 32 x & = 7 (4 x - 5) \\ 27 x & = 28 x - 35 \\ 27 x - 28 x & = - 35 \\ - x & = - 35 \\ x & = 35 \end{matrix}$
$\begin{matrix} \frac{2 x - 9}{10} + \frac{2 x - 3}{2 x - 1} & = \frac{x}{5} \\ \frac{(2 x - 9) (2 x - 1) + 10 (2 x - 3)}{(2 x - 1)} & = \frac{x}{5} \\ (2 x - 9) (2 x - 1) + 10 (2 x - 3) & = 2 x (2 x - 1) \\ 4 x^{2} - 2 x - 18 x + 9 + 20 x - 30 & = 4 x^{2} - 2 x \\ - 21 & = - 2 x \\ x & = \frac{21}{2} \\ x & = 10 \frac{1}{2} \end{matrix}$
$\begin{matrix} \frac{{(3 x - 1)}^{2}}{x - 1} & = \frac{18 x - 1}{2} \\ 2 (9 x^{2} - 6 x + 1) & = (x - 1) (18 x - 1) \\ 18 x^{2} - 12 x + 2 & = 18 x^{2} - x - 18 x + 1 \\ - 12 x & = - 19 x + 1 - 2 \\ 19 x - 12 x & = - 1 \\ 7 x & = - 1 \\ x & = - \frac{1}{7} \end{matrix}$
$\begin{matrix} \frac{2 x + 7}{5 x + 2} - \frac{2 x - 1}{5 x - 4} & = 0 \\ \frac{(2 x + 7) (5 x - 4) - (2 x - 1) (5 x + 2)}{(5 x + 2) (5 x - 4)} & = 0 \\ 10 x^{2} - 8 x + 35 x - 28 - (10 x^{2} + 4 x - 5 x - 2) & = 0 \\ 10 x^{2} + 27 x - 28 - 10 x^{2} + x + 2 & = 0 \\ 28 x - 26 & = 0 \\ 28 x & = 26 \\ x & = \frac{}{} \\ x & = \frac{13}{14} \end{matrix}$
$\begin{matrix} \frac{(5 x - 2) (7 x + 3)}{7 x (5 x - 1)} - 1 & = 0 \\ \frac{(5 x - 2) (7 x + 3)}{7 x (5 x - 1)} & = 1 \\ (5 x - 2) (7 x + 3) & = 7 x (5 x - 1) \\ 35 x^{2} + 15 x - 14 x - 6 & = 35 x^{2} - 7 x \\ x + 7 x & = 6 \\ 8 x & = 6 \\ x & = \frac{}{} \\ x & = \frac{3}{4} \end{matrix}$
$\begin{matrix} \frac{3}{x - 4} & = \frac{2}{x - 3} + \frac{8}{x^{2} - 7 x + 12} \\ \frac{3}{x - 4} & - \frac{2}{x - 3} = \frac{8}{x^{2} - 7 x + 12} \\ \frac{3 (x - 3) - 2 (x - 4)}{(x - 4) (x - 3)} & = \frac{8}{(x - 4) (x - 3)} \\ 3 x - 9 - 2 x + 8 & = 8 \\ x & = 9 \end{matrix}$
$\begin{matrix} \frac{6 x - 1}{18} - \frac{3 (x + 2)}{5 x - 6} & = \frac{1 + 3 x}{9} \\ \frac{(6 x - 1) (5 x - 6) - 54 (x + 2)}{(5 x - 6)} & = \frac{1 + 3 x}{9} \\ \frac{30 x^{2} - 36 x - 5 x + 6 - 54 x - 108}{2 (5 x - 6)} & = 1 + 3 x \\ 30 x^{2} - 95 x - 102 & = 2 (5 x - 6) (1 + 3 x) \\ 30 x^{2} - 95 x - 102 & = 2 (5 x + 15 x^{2} - 6 - 18 x) \\ 30 x^{2} - 95 x - 102 & = 2 (15 x^{2} - 13 x - 6) \\ 30 x^{2} - 95 x - 102 & = 30 x^{2} - 26 x - 12 \\ - 95 x + 26 x & = 102 - 12 \\ - 69 x & = 90 \\ x & = - \frac{}{} \\ x & = - \frac{30}{23} \leftrightarrow - 1 \frac{7}{23} \end{matrix}$
$\begin{matrix} \frac{5}{1 + x} - \frac{3}{1 - x} - \frac{6}{1 - x^{2}} & = 0 \\ \frac{5}{1 + x} - \frac{3}{1 - x} & = \frac{6}{1 - x^{2}} \\ \frac{5 (1 - x) - 3 (1 + x)}{(1 - x) (1 + x)} & = \frac{6}{1 - x^{2}} \\ 5 - 5 x - 3 - 3 x & = 6 \\ 2 - 8 x & = 6 \\ - 8 x & = 6 - 2 \\ - 8 x & = 4 \\ x & = - \frac{4}{} \\ x & = - \frac{1}{2} \end{matrix}$
$\begin{matrix} \frac{1 + 2 x}{1 + 3 x} - \frac{1 - 2 x}{1 - 3 x} & = - \frac{3 x - 14}{1 - 9 x^{2}} \\ \frac{(1 + 2 x) (1 - 3 x) - (1 - 2 x) (1 + 3 x)}{(1 + 3 x) (1 - 3 x)} & = - \frac{3 x - 14}{1 - 9 x^{2}} \\ 1 - 3 x + 2 x - 6 x^{2} - (1 + 3 x - 2 x - 6 x^{2}) & = - 3 x + 14 \\ 1 - x - 6 x^{2} - 1 - x + 6 x^{2} & = - 3 x + 14 \\ - 2 x & = - 3 x + 14 \\ - 2 x + 3 x & = 14 \\ x & = 14 \end{matrix}$
$\begin{matrix} \frac{3 x - 1}{x^{2} + 7 x + 12} & = \frac{1}{2 x + 6} + \frac{7}{6 x + 24} \\ \frac{3 x - 1}{(x + 4) (x + 3)} & = \frac{1}{2 (x + 3)} + \frac{7}{6 (x + 4)} \\ \frac{3 x - 1}{(x + 4) (x + 3)} & = \frac{3 (x + 4) + 7 (x + 3)}{6 (x + 4) (x + 3)} \\ 6 (3 x - 1) & = 3 x + 12 + 7 x + 21 \\ 18 x - 6 & = 10 x + 33 \\ 18 x - 10 x & = 33 + 6 \\ 8 x & = 39 \\ x & = \frac{39}{8} \leftrightarrow 4 \frac{7}{8} \end{matrix}$
$\begin{matrix} \frac{1}{{(x - 1)}^{2}} - \frac{3}{2 x - 2} & = - \frac{3}{2 x + 2} \\ \frac{1}{{(x - 1)}^{2}} & = \frac{3}{2 x - 2} - \frac{3}{2 x + 2} \\ \frac{1}{{(x - 1)}^{2}} & = \frac{3}{2 (x - 1)} - \frac{3}{2 (x + 1)} \\ \frac{1}{{(x - 1)}^{2}} & = \frac{3 (x + 1) - 3 (x - 1)}{2 (x - 1) (x + 1)} \\ \frac{1}{x - 1} & = \frac{3 x + 3 - 3 x + 3}{2 (x + 1)} \\ 2 (x + 1) & = 6 (x - 1) \\ 2 x + 2 & = 6 x - 6 \\ 2 x - 6 x & = - 6 - 2 \\ - 4 x & = - 8 \\ x & = \frac{-}{- 4} \\ x & = 2 \end{matrix}$
$\begin{matrix} \frac{5 x + 13}{15} - \frac{4 x + 5}{5 x - 15} & = \frac{x}{3} \\ \frac{5 x + 13}{15} - \frac{4 x + 5}{5 (x - 3)} & = \frac{x}{3} \\ \frac{(x - 3) (5 x + 13) - 3 (4 x + 5)}{(x - 3)} & = \frac{x}{3} \\ \frac{5 x^{2} + 13 x - 15 x - 39 - 12 x - 15}{5 (x - 3)} & = x \\ 5 x^{2} - 14 x - 54 & = 5 x (x - 3) \\ 5 x^{2} - 14 x - 54 & = 5 x^{2} - 15 x \\ 15 x - 14 x & = 54 \\ x & = 54 \end{matrix}$
$\begin{matrix} \frac{2 x - 1}{2 x + 1} - \frac{x - 4}{3 x - 2} & = \frac{2}{3} \\ \frac{(2 x - 1) (3 x - 2) - (x - 4) (2 x + 1)}{(2 x + 1) (3 x - 2)} & = \frac{2}{3} \\ \frac{6 x^{2} - 4 x - 3 x + 2 - (2 x^{2} + x - 8 x - 4)}{(2 x + 1) (3 x - 2)} & = \frac{2}{3} \\ \frac{6 x^{2} - 7 x + 2 - 2 x^{2} + 7 x + 4}{(2 x + 1) (3 x - 2)} & = \frac{2}{3} \\ \frac{4 x^{2} + 6}{(2 x + 1) (3 x - 2)} & = \frac{2}{3} \\ 3. 2 (2 x^{2} + 3) & = 2 (2 x + 1) (3 x - 2) \\ 6 x^{2} + 9 & = 6 x^{2} - 4 x + 3 x - 2 \\ 9 + 2 & = - x \\ 11 & = - x \\ x & = - 11 \end{matrix}$
$\begin{matrix} \frac{4 x + 3}{2 x - 5} - \frac{3 x + 8}{3 x - 7} & = 1 \\ \frac{(4 x + 3) (3 x - 7) - (3 x + 8) (2 x - 5)}{(2 x - 5) (3 x - 7)} & = 1 \\ 12 x^{2} - 28 x + 9 x - 21 - (6 x^{2} - 15 x + 16 x - 40) & = (2 x - 5) (3 x - 7) \\ 12 x^{2} - 19 x - 21 - 6 x^{2} - x + 40 & = 6 x^{2} - 14 x - 15 x + 35 \\ 6 x^{2} - 20 x + 19 & = 6 x^{2} - 29 x + 35 \\ 29 x - 20 x & = 35 - 19 \\ 9 x & = 16 \\ x & = \frac{16}{9} \leftrightarrow 1 \frac{7}{9} \end{matrix}$
$\begin{matrix} \frac{10 x - 7}{15 x + 3} & = \frac{3 x + 8}{12} - \frac{5 x^{2} - 4}{20 x + 4} \\ \frac{10 x - 7}{3 (5 x + 1)} & = \frac{3 x + 8}{12} - \frac{5 x^{2} - 4}{4 (5 x + 1)} \\ \frac{10 x - 7}{3 (5 x + 1)} & = \frac{(5 x + 1) (3 x + 8) - 3 (5 x^{2} - 4)}{(5 x + 1)} \\ 4 (10 x - 7) & = 15 x^{2} + 40 x + 3 x + 8 - 15 x^{2} + 12 \\ 40 x - 28 & = 43 x + 20 \\ 40 x - 43 x & = 20 + 28 \\ - 3 x & = 48 \\ x & = - \frac{}{3} \\ x & = - 16 \end{matrix}$
$\begin{matrix} \frac{4 x - 1}{5} + \frac{x - 2}{2 x - 7} & = \frac{8 x - 3}{10} - 1 \frac{3}{10} \\ \frac{(4 x - 1) (2 x - 7) + 5 (x - 2)}{5 (2 x - 7)} & = \frac{8 x - 3}{10} - \frac{13}{10} \\ \frac{8 x^{2} - 28 x - 2 x + 7 + 5 x - 10}{5 (2 x - 7)} & = \frac{8 x - 3 - 13}{} \\ \frac{8 x^{2} - 25 x - 3}{2 x - 7} & = \frac{8 x - 16}{2} \\ \frac{8 x^{2} - 25 x - 3}{2 x - 7} & = \frac{(x - 2)}{2} \\ 8 x^{2} - 25 x - 3 & = 4 (x - 2) (2 x - 7) \\ 8 x^{2} - 25 x - 3 & = 4 (2 x^{2} - 7 x - 4 x + 14) \\ 8 x^{2} - 25 x - 3 & = 4 (2 x^{2} - 11 x + 14) \\ 8 x^{2} - 25 x - 3 & = 8 x^{2} - 44 x + 56 \\ 44 x - 25 x & = 56 + 3 \\ 19 x & = 59 \\ x & = \frac{59}{19} \leftrightarrow 3 \frac{2}{19} \end{matrix}$
$\begin{matrix} \frac{1}{x - 1} - \frac{2}{x - 2} & = \frac{3}{2 x - 2} - \frac{2 \frac{1}{3}}{2 x - 4} \\ \frac{x - 2 - 2 (x - 1)}{(x - 1) (x - 2)} & = \frac{3}{2 (x - 1)} - \frac{\frac{7}{3}}{2 (x - 2)} \\ \frac{x - 2 - 2 x + 2}{(x - 1) (x - 2)} & = \frac{3}{2 (x - 1)} - \frac{7}{6 (x - 2)} \\ \frac{- x}{(x - 1) (x - 2)} & = \frac{9 (x - 2) - 7 (x - 1)}{6 (x - 1) (x - 2)} \\ - 6 x & = 9 x - 18 - 7 x + 7 \\ - 6 x & = 2 x - 11 \\ - 6 x - 2 x & = - 11 \\ - 8 x & = - 11 \\ x & = \frac{11}{8} \leftrightarrow 1 \frac{3}{8} \end{matrix}$
$\begin{matrix} \frac{1}{x + 3} - \frac{2}{5 x - 20} & = \frac{1 \frac{1}{2}}{3 x - 12} - \frac{2}{x + 3} \\ \frac{1}{x + 3} - \frac{2}{5 (x - 4)} & = \frac{\frac{3}{2}}{3 (x - 4)} - \frac{2}{x + 3} \\ \frac{5 (x - 4) - 2 (x + 3)}{5 (x + 3) (x - 4)} & = \frac{1}{2 (x - 4)} - \frac{2}{x + 3} \\ \frac{5 x - 20 - 2 x - 6}{5 (x - 4) (x + 3)} & = \frac{x + 3 - 4 (x - 4)}{2 (x - 4) (x + 3)} \\ \frac{3 x - 26}{5} & = \frac{x + 3 - 4 x + 16}{2} \\ 2 (3 x - 26) & = 5 (- 3 x + 19) \\ 6 x - 52 & = - 15 x + 95 \\ 6 x + 15 x & = 95 + 52 \\ 21 x & = 147 \\ x & = \frac{}{21} \\ x & = 7 \end{matrix}$
$\begin{matrix} \frac{1}{6 - 2 x} - \frac{4}{5 - 5 x} & = \frac{10}{12 - 4 x} - \frac{3}{10 - 10 x} \\ \frac{1}{2 (3 - x)} - \frac{4}{5 (1 - x)} & = \frac{}{(3 - x)} - \frac{3}{10 (1 - x)} \\ \frac{1}{2 (3 - x)} - \frac{4}{5 (1 - x)} & = \frac{5}{2 (3 - x)} - \frac{3}{10 (1 - x)} \\ \frac{5 (1 - x) - 8 (3 - x)}{10 (3 - x) (1 - x)} & = \frac{25 (1 - x) - 3 (3 - x)}{10 (3 - x) (1 - x)} \\ 5 (1 - x) - 8 (3 - x) & = 25 (1 - x) - 3 (3 - x) \\ 5 (1 - x) - 25 (1 - x) & = 8 (3 - x) - 3 (3 - x) \\ (1 - x) (5 - 25) & = (3 - x) (8 - 3) \\ - 20 (1 - x) & = 5 (3 - x) \\ - 20 + 20 x & = 15 - 5 x \\ 20 x + 5 x & = 15 + 20 \\ 25 x & = 35 \\ x & = \frac{}{} \\ x & = \frac{7}{5} \leftrightarrow 1 \frac{2}{5} \end{matrix}$
$\begin{matrix} \frac{2}{3} - \frac{6 x^{2}}{9 x^{2} - 1} & = \frac{2}{3 x - 1} \\ \frac{2}{3} - \frac{6 x^{2}}{(3 x - 1) (3 x + 1)} & = \frac{2}{3 x - 1} \\ \frac{2 (3 x - 1) (3 x + 1) - 18 x^{2}}{3 (3 x + 1) (3 x - 1)} & = \frac{2}{3 x - 1} \\ \frac{2 (9 x^{2} - 1) - 18 x^{2}}{3 (3 x + 1)} & = 2 \\ 18 x^{2} - 2 - 18 x^{2} & = 6 (3 x + 1) \\ - 2 & = 18 x + 6 \\ - 6 - 2 & = 18 x \\ - 8 & = 18 x \\ x & = - \frac{}{} \\ x & = - \frac{4}{9} \end{matrix}$
$\begin{matrix} \frac{5 x^{2} - 27 x}{5 x + 3} - \frac{1}{x} & = x - 6 \\ \frac{x (5 x^{2} - 27 x) - (5 x + 3)}{x (5 x + 3)} & = x - 6 \\ 5 x^{3} - 27 x^{2} - 5 x - 3 & = x (5 x + 3) (x - 6) \\ 5 x^{3} - 27 x^{2} - 5 x - 3 & = x (5 x^{2} - 30 x + 3 x - 18) \\ 5 x^{3} - 27 x^{2} - 5 x - 3 & = 5 x^{3} - 27 x^{2} - 18 x \\ 18 x - 5 x & = 3 \\ 13 x & = 3 \\ x & = \frac{3}{13} \end{matrix}$
$\begin{matrix} \frac{4 x + 1}{4 x - 1} - \frac{6}{16 x^{2} - 1} & = \frac{4 x - 1}{4 x + 1} \\ \frac{4 x + 1}{4 x - 1} - \frac{6}{(4 x - 1) (4 x + 1)} & = \frac{4 x - 1}{4 x + 1} \\ \frac{{(4 x + 1)}^{2} - 6}{(4 x + 1) (4 x - 1)} & = \frac{4 x - 1}{4 x + 1} \\ 16 x^{2} + 8 x + 1 - 6 & = {(4 x - 1)}^{2} \\ 16 x^{2} + 8 x - 5 & = 16 x^{2} - 8 x + 1 \\ 8 x + 8 x & = 1 + 5 \\ 16 x & = 6 \\ x & = \frac{}{} \\ x & = \frac{3}{8} \end{matrix}$
$\begin{matrix} 3 (\frac{x - 1}{x + 1}) + 2 (\frac{x + 1}{x - 4}) & = \frac{5 x (x - 1)}{x^{2} - 3 x - 4} \\ \frac{3 (x - 1) (x - 4) + 2 {(x + 1)}^{2}}{(x + 1) (x - 4)} & = \frac{5 x (x - 1)}{(x + 1) (x - 4)} \\ 3 (x^{2} - 5 x + 4) + 2 (x^{2} + 2 x + 1) & = 5 x^{2} - 5 x \\ 3 x^{2} - 15 x + 12 + 2 x^{2} + 4 x + 2 & = 5 x^{2} - 5 x \\ 5 x^{2} - 11 x + 14 & = 5 x^{2} - 5 x \\ - 11 x + 5 x & = - 14 \\ - 6 x & = - 14 \\ x & = \frac{-}{-} \\ x & = \frac{7}{3} \leftrightarrow 2 \frac{1}{3} \end{matrix}$
$\begin{matrix} 2 (\frac{x + 2}{x - 2}) - 3 (\frac{x - 2}{2 x + 3}) & = \frac{x^{2} + 78}{2 x^{2} - x - 6} \\ \frac{2 (x + 2) (2 x + 3) - 3 {(x - 2)}^{2}}{(x - 2) (2 x + 3)} & = \frac{x^{2} + 78}{2 x^{2} - 4 x + 3 x - 6} \\ \frac{2 (2 x^{2} + 3 x + 4 x + 6) - 3 (x^{2} - 4 x + 4)}{(x - 2) (2 x + 3)} & = \frac{x^{2} + 78}{2 x (x - 2) + 3 (x - 2)} \\ \frac{4 x^{2} + 14 x + 12 - 3 x^{2} + 12 x - 12}{(x - 2) (2 x + 3)} & = \frac{x^{2} + 78}{(x - 2) (2 x + 3)} \\ x^{2} + 26 x & = x^{2} + 78 \\ x & = \frac{}{26} \\ x & = 3 \end{matrix}$
$\begin{matrix} \frac{1}{x^{2} + 3 x - 28} - \frac{1}{x^{2} + 12 x + 35} & = \frac{3}{x^{2} + x - 20} \\ \frac{1}{(x + 7) (x - 4)} - \frac{1}{(x + 7) (x + 5)} & = \frac{3}{(x + 5) (x - 4)} \\ \frac{x + 5 - x + 4}{(x + 7) (x + 5) (x - 4)} & = \frac{3}{(x + 5) (x - 4)} \\ 9 & = 3 (x + 7) \\ 9 & = 3 x + 21 \\ 9 - 21 & = 3 x \\ - 12 & = 3 x \\ x & = - \frac{}{3} \\ x & = - 4 \end{matrix}$
$\begin{matrix} \frac{x - 2}{x^{2} + 8 x + 7} & = \frac{2 x - 5}{x^{2} - 49} - \frac{x - 2}{x^{2} - 6 x - 7} \\ \frac{x - 2}{(x + 7) (x + 1)} & = \frac{2 x - 5}{(x + 7) (x - 7)} - \frac{x - 2}{(x - 7) (x + 1)} \\ \frac{x - 2}{(x + 7) (x + 1)} & = \frac{(2 x - 5) (x + 1) - (x - 2) (x + 7)}{(x - 7) (x + 7) (x + 1)} \\ (x - 2) (x - 7) & = 2 x^{2} + 2 x - 5 x - 5 - (x^{2} + 5 x - 14) \\ x^{2} - 9 x + 14 & = 2 x^{2} - 3 x - 5 - x^{2} - 5 x + 14 \\ x^{2} - 9 x & = x^{2} - 8 x - 5 \\ - 9 x + 8 x & = - 5 \\ - x & = - 5 \\ x & = 5 \end{matrix}$
$\begin{matrix} \frac{4 x + 5}{15 x^{2} + 7 x - 2} - \frac{2 x + 3}{12 x^{2} - 7 x - 10} - \frac{2 x - 5}{20 x^{2} - 29 x + 5} & = 0 \\ \frac{4 x + 5}{15 x^{2} - 3 x + 10 x - 2} - \frac{2 x + 3}{12 x^{2} - 15 x + 8 x - 10} - \frac{2 x - 5}{20 x^{2} - 4 x - 25 x + 5} & = 0 \\ \frac{4 x + 5}{3 x (5 x - 1) + 2 (5 x - 1)} - \frac{2 x + 3}{3 x (4 x - 5) + 2 (4 x - 5)} - \frac{2 x - 5}{4 x (5 x - 1) - 5 (5 x - 1)} & = 0 \\ \frac{4 x + 5}{(3 x + 2) (5 x - 1)} - \frac{2 x + 3}{(3 x + 2) (4 x - 5)} - \frac{2 x - 5}{(4 x - 5) (5 x - 1)} & = 0 \\ \frac{(4 x + 5) (4 x - 5) - (2 x + 3) (5 x - 1) - (2 x - 5) (3 x + 2)}{(3 x + 2) (5 x - 1) (4 x - 5)} & = 0 \\ 16 x^{2} - 25 - (10 x^{2} - 2 x + 15 x - 3) - (6 x^{2} + 4 x - 15 x - 10) & = 0 \\ 16 x^{2} - 25 - (10 x^{2} + 13 x - 3) - (6 x^{2} - 11 x - 10) & = 0 \\ 16 x^{2} - 25 - 10 x^{2} - 13 x + 3 - 6 x^{2} + 11 x + 10 & = 0 \\ - 2 x - 12 & = 0 \\ - 2 x & = 12 \\ x & = - \frac{}{2} \\ x & = - 6 \end{matrix}$
$\begin{matrix} \frac{7}{2 x + 1} - \frac{3}{x + 4} & = \frac{2}{x + 1} - \frac{3 (x + 1)}{2 x^{2} + 9 x + 4} \\ \frac{7 (x + 4) - 3 (2 x + 1)}{(2 x + 1) (x + 4)} & = \frac{2}{x + 1} - \frac{3 (x + 1)}{2 x^{2} + x + 8 x + 4} \\ \frac{7 x + 28 - 6 x - 3}{(2 x + 1) (x + 4)} & = \frac{2}{x + 1} - \frac{3 (x + 1)}{x (2 x + 1) + 4 (2 x + 1)} \\ \frac{x + 25}{(2 x + 1) (x + 4)} & = \frac{2}{x + 1} - \frac{3 {(x + 1)}^{2}}{(2 x + 1) (x + 4)} \\ \frac{x + 25}{(2 x + 1) (x + 4)} & = \frac{2 (2 x + 1) (x + 4) - 3 (x^{2} + 2 x + 1)}{(x + 1) (2 x + 1) (x + 4)} \\ (x + 25) (x + 1) & = 2 (2 x^{2} + 8 x + x + 4) - 3 x^{2} - 6 x - 3 \\ x^{2} + 26 x + 25 & = 4 x^{2} + 18 x + 8 - 3 x^{2} - 6 x - 3 \\ x^{2} + 26 x + 25 & = x^{2} + 12 x + 5 \\ 26 x - 12 x & = - 25 + 5 \\ 14 x & = - 20 \\ x & = - \frac{}{} \\ x & = - \frac{10}{7} \leftrightarrow - 1 \frac{3}{7} \end{matrix}$
$\begin{matrix} \frac{{(x + 3)}^{2}}{{(x - 3)}^{2}} & = \frac{x - 1}{x + 1} + \frac{2 (7 x + 1)}{x^{2} - 2 x - 3} \\ \frac{{(x + 3)}^{2}}{{(x - 3)}^{2}} & = \frac{x - 1}{x + 1} + \frac{2 (7 x + 1)}{(x - 3) (x + 1)} \\ \frac{{(x + 3)}^{2}}{{(x - 3)}^{2}} & = \frac{(x - 1) (x - 3) + 2 (7 x + 1)}{(x - 3) (x + 1)} \\ \frac{x^{2} + 6 x + 9}{x - 3} & = \frac{x^{2} - 4 x + 3 + 14 x + 2}{x + 1} \\ \frac{x^{2} + 6 x + 9}{x - 3} & = \frac{x^{2} + 10 x + 5}{x + 1} \\ (x^{2} + 6 x + 9) (x + 1) & = (x - 3) (x^{2} + 10 x + 5) \\ x^{3} + 6 x^{2} + 9 x + x^{2} + 6 x + 9 & = x^{3} + 10 x^{2} + 5 x - 3 x^{2} - 30 x - 15 \\ 7 x^{2} + 15 x + 9 & = 7 x^{2} - 25 x - 15 \\ 15 x + 25 x & = - 15 - 9 \\ 40 x & = - 24 \\ x & = - \frac{}{} \\ x & = - \frac{3}{5} \end{matrix}$
$\begin{matrix} \frac{x - 4}{x + 5} - \frac{x + 1}{x - 2} & = - \frac{12 (x + 3)}{{(x + 5)}^{2}} \\ \frac{(x - 4) (x - 2) - (x + 1) (x + 5)}{(x + 5) (x - 2)} & = - \frac{12 (x + 3)}{{(x + 5)}^{2}} \\ \frac{x^{2} - 6 x + 8 - (x^{2} + 6 x + 5)}{x - 2} & = - \frac{12 (x + 3)}{x + 5} \\ \frac{x^{2} - 6 x + 8 - x^{2} - 6 x - 5}{x - 2} & = - \frac{12 (x + 3)}{x + 5} \\ \frac{- 12 x + 3}{x - 2} & = - \frac{12 (x + 3)}{x + 5} \\ (- 12 x + 3) (x + 5) & = (- 12 x - 36) (x - 2) \\ - 12 x^{2} - 60 x + 3 x + 15 & = - 12 x^{2} + 24 x - 36 x + 72 \\ - 57 x & = - 12 x + 72 - 15 \\ - 57 x + 12 x & = 57 \\ - 45 x & = 57 \\ x & = - \frac{}{} \\ x & = - \frac{19}{15} \leftrightarrow - 1 \frac{4}{15} \end{matrix}$
$\begin{matrix} \frac{x - 3}{x - 4} - \frac{x - 2}{x - 3} & = \frac{x + 2}{x + 1} - \frac{x + 3}{x + 2} \\ \frac{{(x - 3)}^{2} - (x - 2) (x - 4)}{(x - 4) (x - 3)} & = \frac{{(x + 2)}^{2} - (x + 3) (x + 1)}{(x + 1) (x + 2)} \\ \frac{x^{2} - 6 x + 9 - (x^{2} - 6 x + 8)}{(x - 4) (x - 3)} & = \frac{x^{2} + 4 x + 4 - (x^{2} + 4 x + 3)}{(x + 1) (x + 2)} \\ \frac{x^{2} - 6 x + 9 - x^{2} + 6 x - 8}{(x - 4) (x - 3)} & = \frac{x^{2} + 4 x + 4 - x^{2} - 4 x - 3}{(x + 1) (x + 2)} \\ \frac{1}{x^{2} - 7 x + 12} & = \frac{1}{x^{2} + 3 x + 2} \\ x^{2} + 3 x + 2 & = x^{2} - 7 x + 12 \\ 3 x + 7 x & = 12 - 2 \\ 10 x & = 10 \\ x & = \frac{10}{10} \\ x & = 1 \end{matrix}$
$\begin{matrix} \frac{x + 6}{x + 2} - \frac{x + 1}{x - 3} & = \frac{x - 5}{x - 1} - \frac{x}{x + 4} \\ \frac{(x + 6) (x - 3) - (x + 1) (x + 2)}{(x + 2) (x - 3)} & = \frac{(x - 5) (x + 4) - x (x - 1)}{(x - 1) (x + 4)} \\ \frac{x^{2} + 3 x - 18 - (x^{2} + 3 x + 2)}{(x + 2) (x - 3)} & = \frac{x^{2} - x - 20 - (x^{2} - x)}{(x - 1) (x + 4)} \\ \frac{x^{2} + 3 x - 18 - x^{2} - 3 x - 2}{x^{2} - x - 6} & = \frac{x^{2} - x - 20 - x^{2} + x}{x^{2} + 3 x - 4} \\ \frac{- 20}{x^{2} - x - 6} & = \frac{- 20}{x^{2} + 3 x - 4} \\ \frac{1}{x^{2} - x - 6} & = \frac{1}{x^{2} + 3 x - 4} \\ x^{2} + 3 x - 4 & = x^{2} - x - 6 \\ 3 x + x & = - 6 + 4 \\ 4 x & = - 2 \\ x & = - \frac{2}{} \\ x & = - \frac{1}{2} \end{matrix}$