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

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

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[text]

   x       2*x       x    
2*4  - 25*5    - 5*10  > 0

$$- 5 \cdot 10^{x} + 2 \cdot 4^{x} - 25 \cdot 5^{2 x} > 0$$

Подробное решение

[TeX]

Дано неравенство:
$$- 5 \cdot 10^{x} + 2 \cdot 4^{x} - 25 \cdot 5^{2 x} > 0$$
Чтобы решить это нер-во - надо сначала решить соотвествующее ур-ние:
$$- 5 \cdot 10^{x} + 2 \cdot 4^{x} - 25 \cdot 5^{2 x} = 0$$
Решаем:
$$x_{1} = -75.0161482956$$
$$x_{2} = -43.0161482956$$
$$x_{3} = -59.0161482956$$
$$x_{4} = -89.0161482956$$
$$x_{5} = -85.0161482956$$
$$x_{6} = -31.016149502$$
$$x_{7} = -107.016148296$$
$$x_{8} = -95.0161482956$$
$$x_{9} = -35.0161483265$$
$$x_{10} = -67.0161482956$$
$$x_{11} = -47.0161482956$$
$$x_{12} = -103.016148296$$
$$x_{13} = -51.0161482956$$
$$x_{14} = -71.0161482956$$
$$x_{15} = -99.0161482956$$
$$x_{16} = -69.0161482956$$
$$x_{17} = -81.0161482956$$
$$x_{18} = -101.016148296$$
$$x_{19} = -39.0161482964$$
$$x_{20} = -21.0281212285$$
$$x_{21} = -87.0161482956$$
$$x_{22} = -115.016148296$$
$$x_{23} = -33.0161484886$$
$$x_{24} = -105.016148296$$
$$x_{25} = -45.0161482956$$
$$x_{26} = -65.0161482956$$
$$x_{27} = -111.016148296$$
$$x_{28} = -41.0161482957$$
$$x_{29} = -83.0161482956$$
$$x_{30} = -73.0161482956$$
$$x_{31} = -53.0161482956$$
$$x_{32} = -79.0161482956$$
$$x_{33} = -93.0161482956$$
$$x_{34} = -23.0180009193$$
$$x_{35} = -55.0161482956$$
$$x_{36} = -57.0161482956$$
$$x_{37} = -77.0161482956$$
$$x_{38} = -113.016148296$$
$$x_{39} = -63.0161482956$$
$$x_{40} = -97.0161482956$$
$$x_{41} = -37.0161483006$$
$$x_{42} = -109.016148296$$
$$x_{43} = -27.0161954288$$
$$x_{44} = -29.0161558359$$
$$x_{45} = -91.0161482956$$
$$x_{46} = -49.0161482956$$
$$x_{47} = -1.75647079737$$
$$x_{48} = -61.0161482956$$
$$x_{49} = -25.0164431309$$
$$x_{1} = -75.0161482956$$
$$x_{2} = -43.0161482956$$
$$x_{3} = -59.0161482956$$
$$x_{4} = -89.0161482956$$
$$x_{5} = -85.0161482956$$
$$x_{6} = -31.016149502$$
$$x_{7} = -107.016148296$$
$$x_{8} = -95.0161482956$$
$$x_{9} = -35.0161483265$$
$$x_{10} = -67.0161482956$$
$$x_{11} = -47.0161482956$$
$$x_{12} = -103.016148296$$
$$x_{13} = -51.0161482956$$
$$x_{14} = -71.0161482956$$
$$x_{15} = -99.0161482956$$
$$x_{16} = -69.0161482956$$
$$x_{17} = -81.0161482956$$
$$x_{18} = -101.016148296$$
$$x_{19} = -39.0161482964$$
$$x_{20} = -21.0281212285$$
$$x_{21} = -87.0161482956$$
$$x_{22} = -115.016148296$$
$$x_{23} = -33.0161484886$$
$$x_{24} = -105.016148296$$
$$x_{25} = -45.0161482956$$
$$x_{26} = -65.0161482956$$
$$x_{27} = -111.016148296$$
$$x_{28} = -41.0161482957$$
$$x_{29} = -83.0161482956$$
$$x_{30} = -73.0161482956$$
$$x_{31} = -53.0161482956$$
$$x_{32} = -79.0161482956$$
$$x_{33} = -93.0161482956$$
$$x_{34} = -23.0180009193$$
$$x_{35} = -55.0161482956$$
$$x_{36} = -57.0161482956$$
$$x_{37} = -77.0161482956$$
$$x_{38} = -113.016148296$$
$$x_{39} = -63.0161482956$$
$$x_{40} = -97.0161482956$$
$$x_{41} = -37.0161483006$$
$$x_{42} = -109.016148296$$
$$x_{43} = -27.0161954288$$
$$x_{44} = -29.0161558359$$
$$x_{45} = -91.0161482956$$
$$x_{46} = -49.0161482956$$
$$x_{47} = -1.75647079737$$
$$x_{48} = -61.0161482956$$
$$x_{49} = -25.0164431309$$
Данные корни
$$x_{22} = -115.016148296$$
$$x_{38} = -113.016148296$$
$$x_{27} = -111.016148296$$
$$x_{42} = -109.016148296$$
$$x_{7} = -107.016148296$$
$$x_{24} = -105.016148296$$
$$x_{12} = -103.016148296$$
$$x_{18} = -101.016148296$$
$$x_{15} = -99.0161482956$$
$$x_{40} = -97.0161482956$$
$$x_{8} = -95.0161482956$$
$$x_{33} = -93.0161482956$$
$$x_{45} = -91.0161482956$$
$$x_{4} = -89.0161482956$$
$$x_{21} = -87.0161482956$$
$$x_{5} = -85.0161482956$$
$$x_{29} = -83.0161482956$$
$$x_{17} = -81.0161482956$$
$$x_{32} = -79.0161482956$$
$$x_{37} = -77.0161482956$$
$$x_{1} = -75.0161482956$$
$$x_{30} = -73.0161482956$$
$$x_{14} = -71.0161482956$$
$$x_{16} = -69.0161482956$$
$$x_{10} = -67.0161482956$$
$$x_{26} = -65.0161482956$$
$$x_{39} = -63.0161482956$$
$$x_{48} = -61.0161482956$$
$$x_{3} = -59.0161482956$$
$$x_{36} = -57.0161482956$$
$$x_{35} = -55.0161482956$$
$$x_{31} = -53.0161482956$$
$$x_{13} = -51.0161482956$$
$$x_{46} = -49.0161482956$$
$$x_{11} = -47.0161482956$$
$$x_{25} = -45.0161482956$$
$$x_{2} = -43.0161482956$$
$$x_{28} = -41.0161482957$$
$$x_{19} = -39.0161482964$$
$$x_{41} = -37.0161483006$$
$$x_{9} = -35.0161483265$$
$$x_{23} = -33.0161484886$$
$$x_{6} = -31.016149502$$
$$x_{44} = -29.0161558359$$
$$x_{43} = -27.0161954288$$
$$x_{49} = -25.0164431309$$
$$x_{34} = -23.0180009193$$
$$x_{20} = -21.0281212285$$
$$x_{47} = -1.75647079737$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} < x_{22}$$
Возьмём например точку
$$x_{0} = x_{22} - \frac{1}{10}$$
=
$$-115.116148296$$
=
$$-115.116148296$$
подставляем в выражение
$$- 5 \cdot 10^{x} + 2 \cdot 4^{x} - 25 \cdot 5^{2 x} > 0$$

   -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:
$$x < -115.016148296$$

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