4^(x-3)>=(1/2)^(-2*x+1) (неравенство)

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

    Решение

    Вы ввели [src]
     x - 3     -1 + 2*x
    4      >= 2        
    4x3(12)2x+14^{x - 3} \geq \left(\frac{1}{2}\right)^{- 2 x + 1}
    Подробное решение
    Дано неравенство:
    4x3(12)2x+14^{x - 3} \geq \left(\frac{1}{2}\right)^{- 2 x + 1}
    Чтобы решить это нер-во - надо сначала решить соотвествующее ур-ние:
    4x3=(12)2x+14^{x - 3} = \left(\frac{1}{2}\right)^{- 2 x + 1}
    Решаем:
    x1=25.0161482956x_{1} = -25.0161482956
    x2=59.0161482956x_{2} = -59.0161482956
    x3=89.0161482956x_{3} = -89.0161482956
    x4=85.0161482956x_{4} = -85.0161482956
    x5=41.0161482956x_{5} = -41.0161482956
    x6=95.0161482956x_{6} = -95.0161482956
    x7=67.0161482956x_{7} = -67.0161482956
    x8=21.0161482956x_{8} = -21.0161482956
    x9=27.0161482956x_{9} = -27.0161482956
    x10=39.0161482956x_{10} = -39.0161482956
    x11=103.016148296x_{11} = -103.016148296
    x12=35.0161482956x_{12} = -35.0161482956
    x13=51.0161482956x_{13} = -51.0161482956
    x14=31.0161482956x_{14} = -31.0161482956
    x15=71.0161482956x_{15} = -71.0161482956
    x16=99.0161482956x_{16} = -99.0161482956
    x17=69.0161482956x_{17} = -69.0161482956
    x18=81.0161482956x_{18} = -81.0161482956
    x19=101.016148296x_{19} = -101.016148296
    x20=87.0161482956x_{20} = -87.0161482956
    x21=115.016148296x_{21} = -115.016148296
    x22=105.016148296x_{22} = -105.016148296
    x23=45.0161482956x_{23} = -45.0161482956
    x24=65.0161482956x_{24} = -65.0161482956
    x25=111.016148296x_{25} = -111.016148296
    x26=107.016148296x_{26} = -107.016148296
    x27=83.0161482956x_{27} = -83.0161482956
    x28=73.0161482956x_{28} = -73.0161482956
    x29=53.0161482956x_{29} = -53.0161482956
    x30=79.0161482956x_{30} = -79.0161482956
    x31=93.0161482956x_{31} = -93.0161482956
    x32=29.0161482956x_{32} = -29.0161482956
    x33=97.0161482956x_{33} = -97.0161482956
    x34=33.0161482956x_{34} = -33.0161482956
    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=43.0161482956x_{40} = -43.0161482956
    x41=23.0161482956x_{41} = -23.0161482956
    x42=109.016148296x_{42} = -109.016148296
    x43=61.0161482956x_{43} = -61.0161482956
    x44=75.0161482956x_{44} = -75.0161482956
    x45=91.0161482956x_{45} = -91.0161482956
    x46=49.0161482956x_{46} = -49.0161482956
    x47=37.0161482956x_{47} = -37.0161482956
    x48=47.0161482956x_{48} = -47.0161482956
    x1=25.0161482956x_{1} = -25.0161482956
    x2=59.0161482956x_{2} = -59.0161482956
    x3=89.0161482956x_{3} = -89.0161482956
    x4=85.0161482956x_{4} = -85.0161482956
    x5=41.0161482956x_{5} = -41.0161482956
    x6=95.0161482956x_{6} = -95.0161482956
    x7=67.0161482956x_{7} = -67.0161482956
    x8=21.0161482956x_{8} = -21.0161482956
    x9=27.0161482956x_{9} = -27.0161482956
    x10=39.0161482956x_{10} = -39.0161482956
    x11=103.016148296x_{11} = -103.016148296
    x12=35.0161482956x_{12} = -35.0161482956
    x13=51.0161482956x_{13} = -51.0161482956
    x14=31.0161482956x_{14} = -31.0161482956
    x15=71.0161482956x_{15} = -71.0161482956
    x16=99.0161482956x_{16} = -99.0161482956
    x17=69.0161482956x_{17} = -69.0161482956
    x18=81.0161482956x_{18} = -81.0161482956
    x19=101.016148296x_{19} = -101.016148296
    x20=87.0161482956x_{20} = -87.0161482956
    x21=115.016148296x_{21} = -115.016148296
    x22=105.016148296x_{22} = -105.016148296
    x23=45.0161482956x_{23} = -45.0161482956
    x24=65.0161482956x_{24} = -65.0161482956
    x25=111.016148296x_{25} = -111.016148296
    x26=107.016148296x_{26} = -107.016148296
    x27=83.0161482956x_{27} = -83.0161482956
    x28=73.0161482956x_{28} = -73.0161482956
    x29=53.0161482956x_{29} = -53.0161482956
    x30=79.0161482956x_{30} = -79.0161482956
    x31=93.0161482956x_{31} = -93.0161482956
    x32=29.0161482956x_{32} = -29.0161482956
    x33=97.0161482956x_{33} = -97.0161482956
    x34=33.0161482956x_{34} = -33.0161482956
    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=43.0161482956x_{40} = -43.0161482956
    x41=23.0161482956x_{41} = -23.0161482956
    x42=109.016148296x_{42} = -109.016148296
    x43=61.0161482956x_{43} = -61.0161482956
    x44=75.0161482956x_{44} = -75.0161482956
    x45=91.0161482956x_{45} = -91.0161482956
    x46=49.0161482956x_{46} = -49.0161482956
    x47=37.0161482956x_{47} = -37.0161482956
    x48=47.0161482956x_{48} = -47.0161482956
    Данные корни
    x21=115.016148296x_{21} = -115.016148296
    x38=113.016148296x_{38} = -113.016148296
    x25=111.016148296x_{25} = -111.016148296
    x42=109.016148296x_{42} = -109.016148296
    x26=107.016148296x_{26} = -107.016148296
    x22=105.016148296x_{22} = -105.016148296
    x11=103.016148296x_{11} = -103.016148296
    x19=101.016148296x_{19} = -101.016148296
    x16=99.0161482956x_{16} = -99.0161482956
    x33=97.0161482956x_{33} = -97.0161482956
    x6=95.0161482956x_{6} = -95.0161482956
    x31=93.0161482956x_{31} = -93.0161482956
    x45=91.0161482956x_{45} = -91.0161482956
    x3=89.0161482956x_{3} = -89.0161482956
    x20=87.0161482956x_{20} = -87.0161482956
    x4=85.0161482956x_{4} = -85.0161482956
    x27=83.0161482956x_{27} = -83.0161482956
    x18=81.0161482956x_{18} = -81.0161482956
    x30=79.0161482956x_{30} = -79.0161482956
    x37=77.0161482956x_{37} = -77.0161482956
    x44=75.0161482956x_{44} = -75.0161482956
    x28=73.0161482956x_{28} = -73.0161482956
    x15=71.0161482956x_{15} = -71.0161482956
    x17=69.0161482956x_{17} = -69.0161482956
    x7=67.0161482956x_{7} = -67.0161482956
    x24=65.0161482956x_{24} = -65.0161482956
    x39=63.0161482956x_{39} = -63.0161482956
    x43=61.0161482956x_{43} = -61.0161482956
    x2=59.0161482956x_{2} = -59.0161482956
    x36=57.0161482956x_{36} = -57.0161482956
    x35=55.0161482956x_{35} = -55.0161482956
    x29=53.0161482956x_{29} = -53.0161482956
    x13=51.0161482956x_{13} = -51.0161482956
    x46=49.0161482956x_{46} = -49.0161482956
    x48=47.0161482956x_{48} = -47.0161482956
    x23=45.0161482956x_{23} = -45.0161482956
    x40=43.0161482956x_{40} = -43.0161482956
    x5=41.0161482956x_{5} = -41.0161482956
    x10=39.0161482956x_{10} = -39.0161482956
    x47=37.0161482956x_{47} = -37.0161482956
    x12=35.0161482956x_{12} = -35.0161482956
    x34=33.0161482956x_{34} = -33.0161482956
    x14=31.0161482956x_{14} = -31.0161482956
    x32=29.0161482956x_{32} = -29.0161482956
    x9=27.0161482956x_{9} = -27.0161482956
    x1=25.0161482956x_{1} = -25.0161482956
    x41=23.0161482956x_{41} = -23.0161482956
    x8=21.0161482956x_{8} = -21.0161482956
    являются точками смены знака неравенства в решениях.
    Сначала определимся со знаком до крайней левой точки:
    x0x21x_{0} \leq x_{21}
    Возьмём например точку
    x0=x21110x_{0} = x_{21} - \frac{1}{10}
    =
    115.116148296-115.116148296
    =
    115.116148296-115.116148296
    подставляем в выражение
    4x3(12)2x+14^{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
    x115.016148296x \leq -115.016148296
    не выполняется
    значит одно из решений нашего неравенства будет при:
    x115.016148296x113.016148296x \geq -115.016148296 \wedge x \leq -113.016148296
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           x21      x38      x25      x42      x26      x22      x11      x19      x16      x33      x6      x31      x45      x3      x20      x4      x27      x18      x30      x37      x44      x28      x15      x17      x7      x24      x39      x43      x2      x36      x35      x29      x13      x46      x48      x23      x40      x5      x10      x47      x12      x34      x14      x32      x9      x1      x41      x8

    Другие решения неравенства будем получать переходом на следующий полюс
    и т.д.
    Ответ:
    x115.016148296x113.016148296x \geq -115.016148296 \wedge x \leq -113.016148296
    x111.016148296x109.016148296x \geq -111.016148296 \wedge x \leq -109.016148296
    x107.016148296x105.016148296x \geq -107.016148296 \wedge x \leq -105.016148296
    x103.016148296x101.016148296x \geq -103.016148296 \wedge x \leq -101.016148296
    x99.0161482956x97.0161482956x \geq -99.0161482956 \wedge x \leq -97.0161482956
    x95.0161482956x93.0161482956x \geq -95.0161482956 \wedge x \leq -93.0161482956
    x91.0161482956x89.0161482956x \geq -91.0161482956 \wedge x \leq -89.0161482956
    x87.0161482956x85.0161482956x \geq -87.0161482956 \wedge x \leq -85.0161482956
    x83.0161482956x81.0161482956x \geq -83.0161482956 \wedge x \leq -81.0161482956
    x79.0161482956x77.0161482956x \geq -79.0161482956 \wedge x \leq -77.0161482956
    x75.0161482956x73.0161482956x \geq -75.0161482956 \wedge x \leq -73.0161482956
    x71.0161482956x69.0161482956x \geq -71.0161482956 \wedge x \leq -69.0161482956
    x67.0161482956x65.0161482956x \geq -67.0161482956 \wedge x \leq -65.0161482956
    x63.0161482956x61.0161482956x \geq -63.0161482956 \wedge x \leq -61.0161482956
    x59.0161482956x57.0161482956x \geq -59.0161482956 \wedge x \leq -57.0161482956
    x55.0161482956x53.0161482956x \geq -55.0161482956 \wedge x \leq -53.0161482956
    x51.0161482956x49.0161482956x \geq -51.0161482956 \wedge x \leq -49.0161482956
    x47.0161482956x45.0161482956x \geq -47.0161482956 \wedge x \leq -45.0161482956
    x43.0161482956x41.0161482956x \geq -43.0161482956 \wedge x \leq -41.0161482956
    x39.0161482956x37.0161482956x \geq -39.0161482956 \wedge x \leq -37.0161482956
    x35.0161482956x33.0161482956x \geq -35.0161482956 \wedge x \leq -33.0161482956
    x31.0161482956x29.0161482956x \geq -31.0161482956 \wedge x \leq -29.0161482956
    x27.0161482956x25.0161482956x \geq -27.0161482956 \wedge x \leq -25.0161482956
    x23.0161482956x21.0161482956x \geq -23.0161482956 \wedge x \leq -21.0161482956
    Решение неравенства на графике
    0-400-350-300-250-200-150-100-500.01.0