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7.5E: Ejercicios

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    Resolver ecuaciones racionales

    En los siguientes ejercicios, resuelve cada ecuación racional.

    1. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)

    Responder

    \(a=10\)

    2. \(\dfrac{6}{3}-\dfrac{2}{d}=\dfrac{4}{9}\)

    3. \(\dfrac{4}{5}+\dfrac{1}{4}=\dfrac{2}{v}\)

    Responder

    \(v=\dfrac{40}{21}\)

    4. \(\dfrac{3}{8}+\dfrac{2}{y}=\dfrac{1}{4}\)

    5. \(1-\dfrac{2}{m}=\dfrac{8}{m^{2}}\)

    Responder

    \(m=-2,\; m=4\)

    6. \(1+\dfrac{4}{n}=\dfrac{21}{n^{2}}\)

    7. \(1+\dfrac{9}{p}=\dfrac{-20}{p^{2}}\)

    Responder

    \(p=-5, \; p=-4\)

    8. \(1-\dfrac{7}{q}=\dfrac{-6}{q^{2}}\)

    9. \(\dfrac{5}{3 v-2}=\dfrac{7}{4 v}\)

    Responder

    \(v=14\)

    10. \(\dfrac{8}{2 w+1}=\dfrac{3}{w}\)

    11. \(\dfrac{3}{x+4}+\dfrac{7}{x-4}=\dfrac{8}{x^{2}-16}\)

    Responder

    \(x=-\dfrac{4}{5}\)

    12. \(\dfrac{5}{y-9}+\dfrac{1}{y+9}=\dfrac{18}{y^{2}-81}\)

    13. \(\dfrac{8}{z-10}-\dfrac{7}{z+10}=\dfrac{5}{z^{2}-100}\)

    Responder

    \(z=-145\)

    14. \(\dfrac{9}{a+11}-\dfrac{6}{a-11}=\dfrac{6}{a^{2}-121}\)

    15. \(\dfrac{-10}{q-2}-\dfrac{7}{q+4}=1\)

    Responder

    \(q=-18, \; q=-1\)

    16. \(\dfrac{2}{s+7}-\dfrac{3}{s-3}=1\)

    17. \(\dfrac{v-10}{v^{2}-5 v+4}=\dfrac{3}{v-1}-\dfrac{6}{v-4}\)

    Responder

    no hay solución

    18. \(\dfrac{w+8}{w^{2}-11 w+28}=\dfrac{5}{w-7}+\dfrac{2}{w-4}\)

    19. \(\dfrac{x-10}{x^{2}+8 x+12}=\dfrac{3}{x+2}+\dfrac{4}{x+6}\)

    Responder

    no hay solución

    20. \(\dfrac{y-5}{y^{2}-4 y-5}=\dfrac{1}{y+1}+\dfrac{1}{y-5}\)

    21. \(\dfrac{b+3}{3 b}+\dfrac{b}{24}=\dfrac{1}{b}\)

    Responder

    \(b=-8\)

    22. \(\dfrac{c+3}{12 c}+\dfrac{c}{36}=\dfrac{1}{4 c}\)

    23. \(\dfrac{d}{d+3}=\dfrac{18}{d^{2}-9}+4\)

    Responder

    \(d=2\)

    24. \(\dfrac{m}{m+5}=\dfrac{50}{m^{2}-25}+6\)

    25. \(\dfrac{n}{n+2}-3=\dfrac{8}{n^{2}-4}\)

    Responder

    \(m=1\)

    26. \(\dfrac{p}{p+7}-8=\dfrac{98}{p^{2}-49}\)

    27. \(\dfrac{q}{3 q-9}-\dfrac{3}{4 q+12}=\dfrac{7 q^{2}+6 q+63}{24 q^{2}-216}\)

    Responder

    no hay solución

    28. \(\dfrac{r}{3 r-15}-\dfrac{1}{4 r+20}=\dfrac{3 r^{2}+17 r+40}{12 r^{2}-300}\)

    29. \(\dfrac{s}{2 s+6}-\dfrac{2}{5 s+5}=\dfrac{5 s^{2}-3 s-7}{10 s^{2}+40 s+30}\)

    Responder

    \(s=\dfrac{5}{4}\)

    30. \(\dfrac{t}{6 t-12}-\dfrac{5}{2 t+10}=\dfrac{t^{2}-23 t+70}{12 t^{2}+36 t-120}\)

    31. \(\dfrac{2}{x^{2}+2 x-8}-\dfrac{1}{x^{2}+9 x+20}=\dfrac{4}{x^{2}+3 x-10}\)

    Responder

    \(x=-\dfrac{4}{3}\)

    32. \(\dfrac{5}{x^{2}+4 x+3}+\dfrac{2}{x^{2}+x-6}=\dfrac{3}{x^{2}-x-2}\)

    33. \(\dfrac{3}{x^{2}-5 x-6}+\dfrac{3}{x^{2}-7 x+6}=\dfrac{6}{x^{2}-1}\)

    Responder

    no hay solución

    34. \(\dfrac{2}{x^{2}+2 x-3}+\dfrac{3}{x^{2}+4 x+3}=\dfrac{6}{x^{2}-1}\)

    Resolver ecuaciones racionales que involucren funciones

    35. Para la función racional, \(f(x)=\dfrac{x-2}{x^{2}+6 x+8}\):

    1. Encuentra el dominio de la función
    2. Resolver \(f(x)=5\)
    3. Encuentra los puntos en la gráfica en este valor de función
    Responder
    1. El dominio es todos los números reales excepto \(x \neq-2\) y \(x \neq-4\)
    2. \(x=-3, x=-\dfrac{14}{5}\)
    3. \((-3,5),\left(-\dfrac{14}{5}, 5\right)\)

    36. Para la función racional, \(f(x)=\dfrac{x+1}{x^{2}-2 x-3}\):

    1. Encuentra el dominio de la función
    2. Resolver \(f(x)=1\)
    3. Encuentra los puntos en la gráfica en este valor de función

    37. Para la función racional, \(f(x)=\dfrac{2-x}{x^{2}-7 x+10}\):

    1. Encuentra el dominio de la función
    2. Resolver \(f(x)=2\)
    3. Encuentra los puntos en la gráfica en este valor de función
    Responder
    1. El dominio es todos los números reales excepto \(x \neq 2\) y \(x \neq 5\)
    2. \(x=\dfrac{9}{2}\)
    3. \(\left(\dfrac{9}{2}, 2\right)\)

    38. Para la función racional, \(f(x)=\dfrac{5-x}{x^{2}+5 x+6}\):

    1. Encuentra el dominio de la función
    2. Resolver \(f(x)=3\)
    3. Encuentra los puntos en la gráfica en este valor de función

    Resolver una ecuación racional para una variable específica

    En los siguientes ejercicios, resuelve:

    39. \(\dfrac{c}{r}=2 \pi \text { for } r\)

    Responder

    \(r=\dfrac{C}{2 \pi}\)

    40. \(\dfrac{I}{r}=P \text { for } r\)

    41. \(\dfrac{v+3}{w-1}=\dfrac{1}{2} \text { for } w\)

    Responder

    \(w=2 v+7\)

    42. \(\dfrac{x+5}{2-y}=\dfrac{4}{3} \text { for } y\)

    43. \(a=\dfrac{b+3}{c-2} \text { for } c\)

    Responder

    \(c=\dfrac{b+3+2 a}{a}\)

    44. \(m=\dfrac{n}{2-n} \text { for } n\)

    45. \(\dfrac{1}{p}+\dfrac{2}{q}=4 \text { for } p\)

    Responder

    \(p=\dfrac{q}{4 q-2}\)

    46. \(\dfrac{3}{s}+\dfrac{1}{t}=2 \text { for } s\)

    47. \(\dfrac{2}{v}+\dfrac{1}{5}=\dfrac{3}{w} \text { for } w\)

    Responder

    \(w=\dfrac{15 v}{10+v}\)

    48. \(\dfrac{6}{x}+\dfrac{2}{3}=\dfrac{1}{y} \text { for } y\)

    49. \(\dfrac{m+3}{n-2}=\dfrac{4}{5} \text { for } n\)

    Responder

    \(n=\dfrac{5 m+23}{4}\)

    50. \(r=\dfrac{s}{3-t} \text { for } t\)

    51. \(\dfrac{E}{c}=m^{2} \text { for } c\)

    Responder

    \(c=\dfrac{E}{m^{2}}\)

    52. \(\dfrac{R}{T}=W \text { for } T\)

    53. \(\dfrac{3}{x}-\dfrac{5}{y}=\dfrac{1}{4} \text { for } y\)

    Responder

    \(y=\dfrac{20 x}{12-x}\)

    54. \(c=\dfrac{2}{a}+\dfrac{b}{5} \text { for } a\)

    Ejercicios de escritura

    55. Tu compañero de clase está teniendo problemas en esta sección. Anota los pasos que usarías para explicar cómo resolver una ecuación racional.

    Responder

    Las respuestas variarán.

    56. Alek piensa que la ecuación \(\dfrac{y}{y+6}=\dfrac{72}{y^{2}-36}+4\) tiene dos soluciones, \(y=-6\) y \(y=4\). Explica por qué Alek se equivoca.

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