4^(x-3)>=(1/2)^(-2*x+1) (неравенство)

Укажите решение неравенства: 4^(x-3)>=(1/2)^(-2*x+1) (множество решений неравенства)

Вы ввели

[TeX]

[pretty]

[text]

 x - 3     -1 + 2*x
4      >= 2        

$$4^{x - 3} \geq \left(\frac{1}{2}\right)^{- 2 x + 1}$$

Подробное решение

[TeX]

Дано неравенство:
$$4^{x - 3} \geq \left(\frac{1}{2}\right)^{- 2 x + 1}$$
Чтобы решить это нер-во - надо сначала решить соотвествующее ур-ние:
$$4^{x - 3} = \left(\frac{1}{2}\right)^{- 2 x + 1}$$
Решаем:
$$x_{1} = -25.0161482956$$
$$x_{2} = -59.0161482956$$
$$x_{3} = -89.0161482956$$
$$x_{4} = -85.0161482956$$
$$x_{5} = -41.0161482956$$
$$x_{6} = -95.0161482956$$
$$x_{7} = -67.0161482956$$
$$x_{8} = -21.0161482956$$
$$x_{9} = -27.0161482956$$
$$x_{10} = -39.0161482956$$
$$x_{11} = -103.016148296$$
$$x_{12} = -35.0161482956$$
$$x_{13} = -51.0161482956$$
$$x_{14} = -31.0161482956$$
$$x_{15} = -71.0161482956$$
$$x_{16} = -99.0161482956$$
$$x_{17} = -69.0161482956$$
$$x_{18} = -81.0161482956$$
$$x_{19} = -101.016148296$$
$$x_{20} = -87.0161482956$$
$$x_{21} = -115.016148296$$
$$x_{22} = -105.016148296$$
$$x_{23} = -45.0161482956$$
$$x_{24} = -65.0161482956$$
$$x_{25} = -111.016148296$$
$$x_{26} = -107.016148296$$
$$x_{27} = -83.0161482956$$
$$x_{28} = -73.0161482956$$
$$x_{29} = -53.0161482956$$
$$x_{30} = -79.0161482956$$
$$x_{31} = -93.0161482956$$
$$x_{32} = -29.0161482956$$
$$x_{33} = -97.0161482956$$
$$x_{34} = -33.0161482956$$
$$x_{35} = -55.0161482956$$
$$x_{36} = -57.0161482956$$
$$x_{37} = -77.0161482956$$
$$x_{38} = -113.016148296$$
$$x_{39} = -63.0161482956$$
$$x_{40} = -43.0161482956$$
$$x_{41} = -23.0161482956$$
$$x_{42} = -109.016148296$$
$$x_{43} = -61.0161482956$$
$$x_{44} = -75.0161482956$$
$$x_{45} = -91.0161482956$$
$$x_{46} = -49.0161482956$$
$$x_{47} = -37.0161482956$$
$$x_{48} = -47.0161482956$$
$$x_{1} = -25.0161482956$$
$$x_{2} = -59.0161482956$$
$$x_{3} = -89.0161482956$$
$$x_{4} = -85.0161482956$$
$$x_{5} = -41.0161482956$$
$$x_{6} = -95.0161482956$$
$$x_{7} = -67.0161482956$$
$$x_{8} = -21.0161482956$$
$$x_{9} = -27.0161482956$$
$$x_{10} = -39.0161482956$$
$$x_{11} = -103.016148296$$
$$x_{12} = -35.0161482956$$
$$x_{13} = -51.0161482956$$
$$x_{14} = -31.0161482956$$
$$x_{15} = -71.0161482956$$
$$x_{16} = -99.0161482956$$
$$x_{17} = -69.0161482956$$
$$x_{18} = -81.0161482956$$
$$x_{19} = -101.016148296$$
$$x_{20} = -87.0161482956$$
$$x_{21} = -115.016148296$$
$$x_{22} = -105.016148296$$
$$x_{23} = -45.0161482956$$
$$x_{24} = -65.0161482956$$
$$x_{25} = -111.016148296$$
$$x_{26} = -107.016148296$$
$$x_{27} = -83.0161482956$$
$$x_{28} = -73.0161482956$$
$$x_{29} = -53.0161482956$$
$$x_{30} = -79.0161482956$$
$$x_{31} = -93.0161482956$$
$$x_{32} = -29.0161482956$$
$$x_{33} = -97.0161482956$$
$$x_{34} = -33.0161482956$$
$$x_{35} = -55.0161482956$$
$$x_{36} = -57.0161482956$$
$$x_{37} = -77.0161482956$$
$$x_{38} = -113.016148296$$
$$x_{39} = -63.0161482956$$
$$x_{40} = -43.0161482956$$
$$x_{41} = -23.0161482956$$
$$x_{42} = -109.016148296$$
$$x_{43} = -61.0161482956$$
$$x_{44} = -75.0161482956$$
$$x_{45} = -91.0161482956$$
$$x_{46} = -49.0161482956$$
$$x_{47} = -37.0161482956$$
$$x_{48} = -47.0161482956$$
Данные корни
$$x_{21} = -115.016148296$$
$$x_{38} = -113.016148296$$
$$x_{25} = -111.016148296$$
$$x_{42} = -109.016148296$$
$$x_{26} = -107.016148296$$
$$x_{22} = -105.016148296$$
$$x_{11} = -103.016148296$$
$$x_{19} = -101.016148296$$
$$x_{16} = -99.0161482956$$
$$x_{33} = -97.0161482956$$
$$x_{6} = -95.0161482956$$
$$x_{31} = -93.0161482956$$
$$x_{45} = -91.0161482956$$
$$x_{3} = -89.0161482956$$
$$x_{20} = -87.0161482956$$
$$x_{4} = -85.0161482956$$
$$x_{27} = -83.0161482956$$
$$x_{18} = -81.0161482956$$
$$x_{30} = -79.0161482956$$
$$x_{37} = -77.0161482956$$
$$x_{44} = -75.0161482956$$
$$x_{28} = -73.0161482956$$
$$x_{15} = -71.0161482956$$
$$x_{17} = -69.0161482956$$
$$x_{7} = -67.0161482956$$
$$x_{24} = -65.0161482956$$
$$x_{39} = -63.0161482956$$
$$x_{43} = -61.0161482956$$
$$x_{2} = -59.0161482956$$
$$x_{36} = -57.0161482956$$
$$x_{35} = -55.0161482956$$
$$x_{29} = -53.0161482956$$
$$x_{13} = -51.0161482956$$
$$x_{46} = -49.0161482956$$
$$x_{48} = -47.0161482956$$
$$x_{23} = -45.0161482956$$
$$x_{40} = -43.0161482956$$
$$x_{5} = -41.0161482956$$
$$x_{10} = -39.0161482956$$
$$x_{47} = -37.0161482956$$
$$x_{12} = -35.0161482956$$
$$x_{34} = -33.0161482956$$
$$x_{14} = -31.0161482956$$
$$x_{32} = -29.0161482956$$
$$x_{9} = -27.0161482956$$
$$x_{1} = -25.0161482956$$
$$x_{41} = -23.0161482956$$
$$x_{8} = -21.0161482956$$
являются точками смены знака неравенства в решениях.
Сначала определимся со знаком до крайней левой точки:
$$x_{0} \leq x_{21}$$
Возьмём например точку
$$x_{0} = x_{21} - \frac{1}{10}$$
=
$$-115.116148296$$
=
$$-115.116148296$$
подставляем в выражение
$$4^{x - 3} \geq \left(\frac{1}{2}\right)^{- 2 x + 1}$$

 -115.116148296 - 3     --2*-115.116148296 - 1
4                   >= 2                      

7.70890655321392e-72 >= 2.46685009702846e-70

но

7.70890655321392e-72 < 2.46685009702846e-70

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

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Другие решения неравенства будем получать переходом на следующий полюс
и т.д.
Ответ:
$$x \geq -115.016148296 \wedge x \leq -113.016148296$$
$$x \geq -111.016148296 \wedge x \leq -109.016148296$$
$$x \geq -107.016148296 \wedge x \leq -105.016148296$$
$$x \geq -103.016148296 \wedge x \leq -101.016148296$$
$$x \geq -99.0161482956 \wedge x \leq -97.0161482956$$
$$x \geq -95.0161482956 \wedge x \leq -93.0161482956$$
$$x \geq -91.0161482956 \wedge x \leq -89.0161482956$$
$$x \geq -87.0161482956 \wedge x \leq -85.0161482956$$
$$x \geq -83.0161482956 \wedge x \leq -81.0161482956$$
$$x \geq -79.0161482956 \wedge x \leq -77.0161482956$$
$$x \geq -75.0161482956 \wedge x \leq -73.0161482956$$
$$x \geq -71.0161482956 \wedge x \leq -69.0161482956$$
$$x \geq -67.0161482956 \wedge x \leq -65.0161482956$$
$$x \geq -63.0161482956 \wedge x \leq -61.0161482956$$
$$x \geq -59.0161482956 \wedge x \leq -57.0161482956$$
$$x \geq -55.0161482956 \wedge x \leq -53.0161482956$$
$$x \geq -51.0161482956 \wedge x \leq -49.0161482956$$
$$x \geq -47.0161482956 \wedge x \leq -45.0161482956$$
$$x \geq -43.0161482956 \wedge x \leq -41.0161482956$$
$$x \geq -39.0161482956 \wedge x \leq -37.0161482956$$
$$x \geq -35.0161482956 \wedge x \leq -33.0161482956$$
$$x \geq -31.0161482956 \wedge x \leq -29.0161482956$$
$$x \geq -27.0161482956 \wedge x \leq -25.0161482956$$
$$x \geq -23.0161482956 \wedge x \leq -21.0161482956$$