25^(x+1)-29*10^x-4^(x+1)>0 (неравенство)

Укажите решение неравенства: 25^(x+1)-29*10^x-4^(x+1)>0 (множество решений неравенства)

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[text]

  x + 1        x    x + 1    
25      - 29*10  - 4      > 0

$$- 4^{x + 1} + - 29 \cdot 10^{x} + 25^{x + 1} > 0$$

Подробное решение

[TeX]

Дано неравенство:
$$- 4^{x + 1} + - 29 \cdot 10^{x} + 25^{x + 1} > 0$$
Чтобы решить это нер-во - надо сначала решить соотвествующее ур-ние:
$$- 4^{x + 1} + - 29 \cdot 10^{x} + 25^{x + 1} = 0$$
Решаем:
$$x_{1} = -43.0161482956$$
$$x_{2} = -59.0161482956$$
$$x_{3} = -89.0161482956$$
$$x_{4} = -85.0161482956$$
$$x_{5} = -107.016148296$$
$$x_{6} = -95.0161482956$$
$$x_{7} = -27.0160116495$$
$$x_{8} = -20.9838018399$$
$$x_{9} = -67.0161482956$$
$$x_{10} = -47.0161482956$$
$$x_{11} = -49.0161482956$$
$$x_{12} = -51.0161482956$$
$$x_{13} = 0.273290551007$$
$$x_{14} = -99.0161482956$$
$$x_{15} = -35.016148206$$
$$x_{16} = -37.0161482813$$
$$x_{17} = -81.0161482956$$
$$x_{18} = -101.016148296$$
$$x_{19} = -23.0108368572$$
$$x_{20} = -87.0161482956$$
$$x_{21} = -115.016148296$$
$$x_{22} = -39.0161482933$$
$$x_{23} = -105.016148296$$
$$x_{24} = -25.015294841$$
$$x_{25} = -69.0161482956$$
$$x_{26} = -45.0161482956$$
$$x_{27} = -65.0161482956$$
$$x_{28} = -111.016148296$$
$$x_{29} = -57.0161482956$$
$$x_{30} = -83.0161482956$$
$$x_{31} = -73.0161482956$$
$$x_{32} = -53.0161482956$$
$$x_{33} = -79.0161482956$$
$$x_{34} = -41.0161482952$$
$$x_{35} = -93.0161482956$$
$$x_{36} = -97.0161482956$$
$$x_{37} = -55.0161482956$$
$$x_{38} = -29.0161264298$$
$$x_{39} = -71.0161482956$$
$$x_{40} = -77.0161482956$$
$$x_{41} = -113.016148296$$
$$x_{42} = -63.0161482956$$
$$x_{43} = -109.016148296$$
$$x_{44} = -61.0161482956$$
$$x_{45} = -75.0161482956$$
$$x_{46} = -91.0161482956$$
$$x_{47} = -103.016148296$$
$$x_{48} = -33.0161477358$$
$$x_{49} = -31.016144797$$
$$x_{1} = -43.0161482956$$
$$x_{2} = -59.0161482956$$
$$x_{3} = -89.0161482956$$
$$x_{4} = -85.0161482956$$
$$x_{5} = -107.016148296$$
$$x_{6} = -95.0161482956$$
$$x_{7} = -27.0160116495$$
$$x_{8} = -20.9838018399$$
$$x_{9} = -67.0161482956$$
$$x_{10} = -47.0161482956$$
$$x_{11} = -49.0161482956$$
$$x_{12} = -51.0161482956$$
$$x_{13} = 0.273290551007$$
$$x_{14} = -99.0161482956$$
$$x_{15} = -35.016148206$$
$$x_{16} = -37.0161482813$$
$$x_{17} = -81.0161482956$$
$$x_{18} = -101.016148296$$
$$x_{19} = -23.0108368572$$
$$x_{20} = -87.0161482956$$
$$x_{21} = -115.016148296$$
$$x_{22} = -39.0161482933$$
$$x_{23} = -105.016148296$$
$$x_{24} = -25.015294841$$
$$x_{25} = -69.0161482956$$
$$x_{26} = -45.0161482956$$
$$x_{27} = -65.0161482956$$
$$x_{28} = -111.016148296$$
$$x_{29} = -57.0161482956$$
$$x_{30} = -83.0161482956$$
$$x_{31} = -73.0161482956$$
$$x_{32} = -53.0161482956$$
$$x_{33} = -79.0161482956$$
$$x_{34} = -41.0161482952$$
$$x_{35} = -93.0161482956$$
$$x_{36} = -97.0161482956$$
$$x_{37} = -55.0161482956$$
$$x_{38} = -29.0161264298$$
$$x_{39} = -71.0161482956$$
$$x_{40} = -77.0161482956$$
$$x_{41} = -113.016148296$$
$$x_{42} = -63.0161482956$$
$$x_{43} = -109.016148296$$
$$x_{44} = -61.0161482956$$
$$x_{45} = -75.0161482956$$
$$x_{46} = -91.0161482956$$
$$x_{47} = -103.016148296$$
$$x_{48} = -33.0161477358$$
$$x_{49} = -31.016144797$$
Данные корни
$$x_{21} = -115.016148296$$
$$x_{41} = -113.016148296$$
$$x_{28} = -111.016148296$$
$$x_{43} = -109.016148296$$
$$x_{5} = -107.016148296$$
$$x_{23} = -105.016148296$$
$$x_{47} = -103.016148296$$
$$x_{18} = -101.016148296$$
$$x_{14} = -99.0161482956$$
$$x_{36} = -97.0161482956$$
$$x_{6} = -95.0161482956$$
$$x_{35} = -93.0161482956$$
$$x_{46} = -91.0161482956$$
$$x_{3} = -89.0161482956$$
$$x_{20} = -87.0161482956$$
$$x_{4} = -85.0161482956$$
$$x_{30} = -83.0161482956$$
$$x_{17} = -81.0161482956$$
$$x_{33} = -79.0161482956$$
$$x_{40} = -77.0161482956$$
$$x_{45} = -75.0161482956$$
$$x_{31} = -73.0161482956$$
$$x_{39} = -71.0161482956$$
$$x_{25} = -69.0161482956$$
$$x_{9} = -67.0161482956$$
$$x_{27} = -65.0161482956$$
$$x_{42} = -63.0161482956$$
$$x_{44} = -61.0161482956$$
$$x_{2} = -59.0161482956$$
$$x_{29} = -57.0161482956$$
$$x_{37} = -55.0161482956$$
$$x_{32} = -53.0161482956$$
$$x_{12} = -51.0161482956$$
$$x_{11} = -49.0161482956$$
$$x_{10} = -47.0161482956$$
$$x_{26} = -45.0161482956$$
$$x_{1} = -43.0161482956$$
$$x_{34} = -41.0161482952$$
$$x_{22} = -39.0161482933$$
$$x_{16} = -37.0161482813$$
$$x_{15} = -35.016148206$$
$$x_{48} = -33.0161477358$$
$$x_{49} = -31.016144797$$
$$x_{38} = -29.0161264298$$
$$x_{7} = -27.0160116495$$
$$x_{24} = -25.015294841$$
$$x_{19} = -23.0108368572$$
$$x_{8} = -20.9838018399$$
$$x_{13} = 0.273290551007$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} < x_{21}$$
Возьмём например точку
$$x_{0} = x_{21} - \frac{1}{10}$$
=
$$-115.116148296$$
=
$$-115.116148296$$
подставляем в выражение
$$- 4^{x + 1} + - 29 \cdot 10^{x} + 25^{x + 1} > 0$$

  -115.116148296 + 1        -115.116148296    -115.116148296 + 1    
25                   - 29*10               - 4                   > 0

-1.97348007762276e-69 > 0

Тогда
$$x < -115.016148296$$
не выполняется
значит одно из решений нашего неравенства будет при:
$$x > -115.016148296 \wedge x < -113.016148296$$

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Другие решения неравенства будем получать переходом на следующий полюс
и т.д.
Ответ:
$$x > -115.016148296 \wedge x < -113.016148296$$
$$x > -111.016148296 \wedge x < -109.016148296$$
$$x > -107.016148296 \wedge x < -105.016148296$$
$$x > -103.016148296 \wedge x < -101.016148296$$
$$x > -99.0161482956 \wedge x < -97.0161482956$$
$$x > -95.0161482956 \wedge x < -93.0161482956$$
$$x > -91.0161482956 \wedge x < -89.0161482956$$
$$x > -87.0161482956 \wedge x < -85.0161482956$$
$$x > -83.0161482956 \wedge x < -81.0161482956$$
$$x > -79.0161482956 \wedge x < -77.0161482956$$
$$x > -75.0161482956 \wedge x < -73.0161482956$$
$$x > -71.0161482956 \wedge x < -69.0161482956$$
$$x > -67.0161482956 \wedge x < -65.0161482956$$
$$x > -63.0161482956 \wedge x < -61.0161482956$$
$$x > -59.0161482956 \wedge x < -57.0161482956$$
$$x > -55.0161482956 \wedge x < -53.0161482956$$
$$x > -51.0161482956 \wedge x < -49.0161482956$$
$$x > -47.0161482956 \wedge x < -45.0161482956$$
$$x > -43.0161482956 \wedge x < -41.0161482952$$
$$x > -39.0161482933 \wedge x < -37.0161482813$$
$$x > -35.016148206 \wedge x < -33.0161477358$$
$$x > -31.016144797 \wedge x < -29.0161264298$$
$$x > -27.0160116495 \wedge x < -25.015294841$$
$$x > -23.0108368572 \wedge x < -20.9838018399$$
$$x > 0.273290551007$$