2*4^x-25*5^(2*x)-5*10^x>0 (неравенство)

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    Укажите решение неравенства: 2*4^x-25*5^(2*x)-5*10^x>0 (множество решений неравенства)

    Решение

    Вы ввели [src]
       x       2*x       x    
    2*4  - 25*5    - 5*10  > 0
    510x+24x2552x>0- 5 \cdot 10^{x} + 2 \cdot 4^{x} - 25 \cdot 5^{2 x} > 0
    Подробное решение
    Дано неравенство:
    510x+24x2552x>0- 5 \cdot 10^{x} + 2 \cdot 4^{x} - 25 \cdot 5^{2 x} > 0
    Чтобы решить это нер-во - надо сначала решить соотвествующее ур-ние:
    510x+24x2552x=0- 5 \cdot 10^{x} + 2 \cdot 4^{x} - 25 \cdot 5^{2 x} = 0
    Решаем:
    x1=75.0161482956x_{1} = -75.0161482956
    x2=43.0161482956x_{2} = -43.0161482956
    x3=59.0161482956x_{3} = -59.0161482956
    x4=89.0161482956x_{4} = -89.0161482956
    x5=85.0161482956x_{5} = -85.0161482956
    x6=31.016149502x_{6} = -31.016149502
    x7=107.016148296x_{7} = -107.016148296
    x8=95.0161482956x_{8} = -95.0161482956
    x9=35.0161483265x_{9} = -35.0161483265
    x10=67.0161482956x_{10} = -67.0161482956
    x11=47.0161482956x_{11} = -47.0161482956
    x12=103.016148296x_{12} = -103.016148296
    x13=51.0161482956x_{13} = -51.0161482956
    x14=71.0161482956x_{14} = -71.0161482956
    x15=99.0161482956x_{15} = -99.0161482956
    x16=69.0161482956x_{16} = -69.0161482956
    x17=81.0161482956x_{17} = -81.0161482956
    x18=101.016148296x_{18} = -101.016148296
    x19=39.0161482964x_{19} = -39.0161482964
    x20=21.0281212285x_{20} = -21.0281212285
    x21=87.0161482956x_{21} = -87.0161482956
    x22=115.016148296x_{22} = -115.016148296
    x23=33.0161484886x_{23} = -33.0161484886
    x24=105.016148296x_{24} = -105.016148296
    x25=45.0161482956x_{25} = -45.0161482956
    x26=65.0161482956x_{26} = -65.0161482956
    x27=111.016148296x_{27} = -111.016148296
    x28=41.0161482957x_{28} = -41.0161482957
    x29=83.0161482956x_{29} = -83.0161482956
    x30=73.0161482956x_{30} = -73.0161482956
    x31=53.0161482956x_{31} = -53.0161482956
    x32=79.0161482956x_{32} = -79.0161482956
    x33=93.0161482956x_{33} = -93.0161482956
    x34=23.0180009193x_{34} = -23.0180009193
    x35=55.0161482956x_{35} = -55.0161482956
    x36=57.0161482956x_{36} = -57.0161482956
    x37=77.0161482956x_{37} = -77.0161482956
    x38=113.016148296x_{38} = -113.016148296
    x39=63.0161482956x_{39} = -63.0161482956
    x40=97.0161482956x_{40} = -97.0161482956
    x41=37.0161483006x_{41} = -37.0161483006
    x42=109.016148296x_{42} = -109.016148296
    x43=27.0161954288x_{43} = -27.0161954288
    x44=29.0161558359x_{44} = -29.0161558359
    x45=91.0161482956x_{45} = -91.0161482956
    x46=49.0161482956x_{46} = -49.0161482956
    x47=1.75647079737x_{47} = -1.75647079737
    x48=61.0161482956x_{48} = -61.0161482956
    x49=25.0164431309x_{49} = -25.0164431309
    x1=75.0161482956x_{1} = -75.0161482956
    x2=43.0161482956x_{2} = -43.0161482956
    x3=59.0161482956x_{3} = -59.0161482956
    x4=89.0161482956x_{4} = -89.0161482956
    x5=85.0161482956x_{5} = -85.0161482956
    x6=31.016149502x_{6} = -31.016149502
    x7=107.016148296x_{7} = -107.016148296
    x8=95.0161482956x_{8} = -95.0161482956
    x9=35.0161483265x_{9} = -35.0161483265
    x10=67.0161482956x_{10} = -67.0161482956
    x11=47.0161482956x_{11} = -47.0161482956
    x12=103.016148296x_{12} = -103.016148296
    x13=51.0161482956x_{13} = -51.0161482956
    x14=71.0161482956x_{14} = -71.0161482956
    x15=99.0161482956x_{15} = -99.0161482956
    x16=69.0161482956x_{16} = -69.0161482956
    x17=81.0161482956x_{17} = -81.0161482956
    x18=101.016148296x_{18} = -101.016148296
    x19=39.0161482964x_{19} = -39.0161482964
    x20=21.0281212285x_{20} = -21.0281212285
    x21=87.0161482956x_{21} = -87.0161482956
    x22=115.016148296x_{22} = -115.016148296
    x23=33.0161484886x_{23} = -33.0161484886
    x24=105.016148296x_{24} = -105.016148296
    x25=45.0161482956x_{25} = -45.0161482956
    x26=65.0161482956x_{26} = -65.0161482956
    x27=111.016148296x_{27} = -111.016148296
    x28=41.0161482957x_{28} = -41.0161482957
    x29=83.0161482956x_{29} = -83.0161482956
    x30=73.0161482956x_{30} = -73.0161482956
    x31=53.0161482956x_{31} = -53.0161482956
    x32=79.0161482956x_{32} = -79.0161482956
    x33=93.0161482956x_{33} = -93.0161482956
    x34=23.0180009193x_{34} = -23.0180009193
    x35=55.0161482956x_{35} = -55.0161482956
    x36=57.0161482956x_{36} = -57.0161482956
    x37=77.0161482956x_{37} = -77.0161482956
    x38=113.016148296x_{38} = -113.016148296
    x39=63.0161482956x_{39} = -63.0161482956
    x40=97.0161482956x_{40} = -97.0161482956
    x41=37.0161483006x_{41} = -37.0161483006
    x42=109.016148296x_{42} = -109.016148296
    x43=27.0161954288x_{43} = -27.0161954288
    x44=29.0161558359x_{44} = -29.0161558359
    x45=91.0161482956x_{45} = -91.0161482956
    x46=49.0161482956x_{46} = -49.0161482956
    x47=1.75647079737x_{47} = -1.75647079737
    x48=61.0161482956x_{48} = -61.0161482956
    x49=25.0164431309x_{49} = -25.0164431309
    Данные корни
    x22=115.016148296x_{22} = -115.016148296
    x38=113.016148296x_{38} = -113.016148296
    x27=111.016148296x_{27} = -111.016148296
    x42=109.016148296x_{42} = -109.016148296
    x7=107.016148296x_{7} = -107.016148296
    x24=105.016148296x_{24} = -105.016148296
    x12=103.016148296x_{12} = -103.016148296
    x18=101.016148296x_{18} = -101.016148296
    x15=99.0161482956x_{15} = -99.0161482956
    x40=97.0161482956x_{40} = -97.0161482956
    x8=95.0161482956x_{8} = -95.0161482956
    x33=93.0161482956x_{33} = -93.0161482956
    x45=91.0161482956x_{45} = -91.0161482956
    x4=89.0161482956x_{4} = -89.0161482956
    x21=87.0161482956x_{21} = -87.0161482956
    x5=85.0161482956x_{5} = -85.0161482956
    x29=83.0161482956x_{29} = -83.0161482956
    x17=81.0161482956x_{17} = -81.0161482956
    x32=79.0161482956x_{32} = -79.0161482956
    x37=77.0161482956x_{37} = -77.0161482956
    x1=75.0161482956x_{1} = -75.0161482956
    x30=73.0161482956x_{30} = -73.0161482956
    x14=71.0161482956x_{14} = -71.0161482956
    x16=69.0161482956x_{16} = -69.0161482956
    x10=67.0161482956x_{10} = -67.0161482956
    x26=65.0161482956x_{26} = -65.0161482956
    x39=63.0161482956x_{39} = -63.0161482956
    x48=61.0161482956x_{48} = -61.0161482956
    x3=59.0161482956x_{3} = -59.0161482956
    x36=57.0161482956x_{36} = -57.0161482956
    x35=55.0161482956x_{35} = -55.0161482956
    x31=53.0161482956x_{31} = -53.0161482956
    x13=51.0161482956x_{13} = -51.0161482956
    x46=49.0161482956x_{46} = -49.0161482956
    x11=47.0161482956x_{11} = -47.0161482956
    x25=45.0161482956x_{25} = -45.0161482956
    x2=43.0161482956x_{2} = -43.0161482956
    x28=41.0161482957x_{28} = -41.0161482957
    x19=39.0161482964x_{19} = -39.0161482964
    x41=37.0161483006x_{41} = -37.0161483006
    x9=35.0161483265x_{9} = -35.0161483265
    x23=33.0161484886x_{23} = -33.0161484886
    x6=31.016149502x_{6} = -31.016149502
    x44=29.0161558359x_{44} = -29.0161558359
    x43=27.0161954288x_{43} = -27.0161954288
    x49=25.0164431309x_{49} = -25.0164431309
    x34=23.0180009193x_{34} = -23.0180009193
    x20=21.0281212285x_{20} = -21.0281212285
    x47=1.75647079737x_{47} = -1.75647079737
    являются точками смены знака неравенства в решениях.
    Сначала определимся со знаком до крайней левой точки:
    x0<x22x_{0} < x_{22}
    Возьмём например точку
    x0=x22110x_{0} = x_{22} - \frac{1}{10}
    =
    115.116148296-115.116148296
    =
    115.116148296-115.116148296
    подставляем в выражение
    510x+24x2552x>0- 5 \cdot 10^{x} + 2 \cdot 4^{x} - 25 \cdot 5^{2 x} > 0
       -115.116148296       2*-115.116148296       -115.116148296    
    2*4               - 25*5                 - 5*10               > 0

    9.86740038811382e-70 > 0

    значит одно из решений нашего неравенства будет при:
    x<115.016148296x < -115.016148296
     _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____           _____          
          \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \         /     \    
    -------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
           x22      x38      x27      x42      x7      x24      x12      x18      x15      x40      x8      x33      x45      x4      x21      x5      x29      x17      x32      x37      x1      x30      x14      x16      x10      x26      x39      x48      x3      x36      x35      x31      x13      x46      x11      x25      x2      x28      x19      x41      x9      x23      x6      x44      x43      x49      x34      x20      x47

    Другие решения неравенства будем получать переходом на следующий полюс
    и т.д.
    Ответ:
    x<115.016148296x < -115.016148296
    x>113.016148296x<111.016148296x > -113.016148296 \wedge x < -111.016148296
    x>109.016148296x<107.016148296x > -109.016148296 \wedge x < -107.016148296
    x>105.016148296x<103.016148296x > -105.016148296 \wedge x < -103.016148296
    x>101.016148296x<99.0161482956x > -101.016148296 \wedge x < -99.0161482956
    x>97.0161482956x<95.0161482956x > -97.0161482956 \wedge x < -95.0161482956
    x>93.0161482956x<91.0161482956x > -93.0161482956 \wedge x < -91.0161482956
    x>89.0161482956x<87.0161482956x > -89.0161482956 \wedge x < -87.0161482956
    x>85.0161482956x<83.0161482956x > -85.0161482956 \wedge x < -83.0161482956
    x>81.0161482956x<79.0161482956x > -81.0161482956 \wedge x < -79.0161482956
    x>77.0161482956x<75.0161482956x > -77.0161482956 \wedge x < -75.0161482956
    x>73.0161482956x<71.0161482956x > -73.0161482956 \wedge x < -71.0161482956
    x>69.0161482956x<67.0161482956x > -69.0161482956 \wedge x < -67.0161482956
    x>65.0161482956x<63.0161482956x > -65.0161482956 \wedge x < -63.0161482956
    x>61.0161482956x<59.0161482956x > -61.0161482956 \wedge x < -59.0161482956
    x>57.0161482956x<55.0161482956x > -57.0161482956 \wedge x < -55.0161482956
    x>53.0161482956x<51.0161482956x > -53.0161482956 \wedge x < -51.0161482956
    x>49.0161482956x<47.0161482956x > -49.0161482956 \wedge x < -47.0161482956
    x>45.0161482956x<43.0161482956x > -45.0161482956 \wedge x < -43.0161482956
    x>41.0161482957x<39.0161482964x > -41.0161482957 \wedge x < -39.0161482964
    x>37.0161483006x<35.0161483265x > -37.0161483006 \wedge x < -35.0161483265
    x>33.0161484886x<31.016149502x > -33.0161484886 \wedge x < -31.016149502
    x>29.0161558359x<27.0161954288x > -29.0161558359 \wedge x < -27.0161954288
    x>25.0164431309x<23.0180009193x > -25.0164431309 \wedge x < -23.0180009193
    x>21.0281212285x<1.75647079737x > -21.0281212285 \wedge x < -1.75647079737
    Решение неравенства на графике
    501234-7-6-5-4-3-2-1-2020