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2*20^x-17*10^x-2*8^x+8*5^x+17*4^x-2^(x-3)<=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
    x        x      x      x       x    x - 3     
2*20  - 17*10  - 2*8  + 8*5  + 17*4  - 2      <= 0
$$- 2^{x - 3} + \left(17 \cdot 4^{x} + \left(8 \cdot 5^{x} + \left(- 2 \cdot 8^{x} + \left(- 17 \cdot 10^{x} + 2 \cdot 20^{x}\right)\right)\right)\right) \leq 0$$
-2^(x - 3) + 17*4^x + 8*5^x - 2*8^x - 17*10^x + 2*20^x <= 0
Solución detallada
Se da la desigualdad:
$$- 2^{x - 3} + \left(17 \cdot 4^{x} + \left(8 \cdot 5^{x} + \left(- 2 \cdot 8^{x} + \left(- 17 \cdot 10^{x} + 2 \cdot 20^{x}\right)\right)\right)\right) \leq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$- 2^{x - 3} + \left(17 \cdot 4^{x} + \left(8 \cdot 5^{x} + \left(- 2 \cdot 8^{x} + \left(- 17 \cdot 10^{x} + 2 \cdot 20^{x}\right)\right)\right)\right) = 0$$
Resolvemos:
$$x_{1} = -118.176760093132$$
$$x_{2} = 2.99343804847255$$
$$x_{3} = -90.176760093132$$
$$x_{4} = -106.176760093132$$
$$x_{5} = -126.176760093132$$
$$x_{6} = -76.1767600931369$$
$$x_{7} = -64.1767601130287$$
$$x_{8} = -110.176760093132$$
$$x_{9} = -40.6038236940885$$
$$x_{10} = -96.176760093132$$
$$x_{11} = -114.176760093132$$
$$x_{12} = -124.176760093132$$
$$x_{13} = -116.176760093132$$
$$x_{14} = -128.176760093132$$
$$x_{15} = -120.176760093132$$
$$x_{16} = -122.176760093132$$
$$x_{17} = -56.1767651937535$$
$$x_{18} = -70.1767600934429$$
$$x_{19} = -58.1767613675219$$
$$x_{20} = -78.1767600931332$$
$$x_{21} = -50.1770881070134$$
$$x_{22} = -68.1767600943754$$
$$x_{23} = -42.2668361314626$$
$$x_{24} = -60.1767604116072$$
$$x_{25} = -44.1983252994765$$
$$x_{26} = -62.1767601727313$$
$$x_{27} = -108.176760093132$$
$$x_{28} = 0.535347584231804$$
$$x_{29} = -94.176760093132$$
$$x_{30} = -80.1767600931323$$
$$x_{31} = -100.176760093132$$
$$x_{32} = -130.176760093132$$
$$x_{33} = -84.176760093132$$
$$x_{34} = -7.21435991128888$$
$$x_{35} = -86.176760093132$$
$$x_{36} = -52.1768419014221$$
$$x_{37} = -98.176760093132$$
$$x_{38} = -74.1767600931515$$
$$x_{39} = -82.1767600931321$$
$$x_{40} = -112.176760093132$$
$$x_{41} = -102.176760093132$$
$$x_{42} = -72.1767600932097$$
$$x_{43} = -88.176760093132$$
$$x_{44} = -48.1780773770746$$
$$x_{45} = -104.176760093132$$
$$x_{46} = -66.1767600981057$$
$$x_{47} = -54.1767805148382$$
$$x_{48} = -92.176760093132$$
$$x_{49} = -46.1820676051361$$
$$x_{1} = -118.176760093132$$
$$x_{2} = 2.99343804847255$$
$$x_{3} = -90.176760093132$$
$$x_{4} = -106.176760093132$$
$$x_{5} = -126.176760093132$$
$$x_{6} = -76.1767600931369$$
$$x_{7} = -64.1767601130287$$
$$x_{8} = -110.176760093132$$
$$x_{9} = -40.6038236940885$$
$$x_{10} = -96.176760093132$$
$$x_{11} = -114.176760093132$$
$$x_{12} = -124.176760093132$$
$$x_{13} = -116.176760093132$$
$$x_{14} = -128.176760093132$$
$$x_{15} = -120.176760093132$$
$$x_{16} = -122.176760093132$$
$$x_{17} = -56.1767651937535$$
$$x_{18} = -70.1767600934429$$
$$x_{19} = -58.1767613675219$$
$$x_{20} = -78.1767600931332$$
$$x_{21} = -50.1770881070134$$
$$x_{22} = -68.1767600943754$$
$$x_{23} = -42.2668361314626$$
$$x_{24} = -60.1767604116072$$
$$x_{25} = -44.1983252994765$$
$$x_{26} = -62.1767601727313$$
$$x_{27} = -108.176760093132$$
$$x_{28} = 0.535347584231804$$
$$x_{29} = -94.176760093132$$
$$x_{30} = -80.1767600931323$$
$$x_{31} = -100.176760093132$$
$$x_{32} = -130.176760093132$$
$$x_{33} = -84.176760093132$$
$$x_{34} = -7.21435991128888$$
$$x_{35} = -86.176760093132$$
$$x_{36} = -52.1768419014221$$
$$x_{37} = -98.176760093132$$
$$x_{38} = -74.1767600931515$$
$$x_{39} = -82.1767600931321$$
$$x_{40} = -112.176760093132$$
$$x_{41} = -102.176760093132$$
$$x_{42} = -72.1767600932097$$
$$x_{43} = -88.176760093132$$
$$x_{44} = -48.1780773770746$$
$$x_{45} = -104.176760093132$$
$$x_{46} = -66.1767600981057$$
$$x_{47} = -54.1767805148382$$
$$x_{48} = -92.176760093132$$
$$x_{49} = -46.1820676051361$$
Las raíces dadas
$$x_{32} = -130.176760093132$$
$$x_{14} = -128.176760093132$$
$$x_{5} = -126.176760093132$$
$$x_{12} = -124.176760093132$$
$$x_{16} = -122.176760093132$$
$$x_{15} = -120.176760093132$$
$$x_{1} = -118.176760093132$$
$$x_{13} = -116.176760093132$$
$$x_{11} = -114.176760093132$$
$$x_{40} = -112.176760093132$$
$$x_{8} = -110.176760093132$$
$$x_{27} = -108.176760093132$$
$$x_{4} = -106.176760093132$$
$$x_{45} = -104.176760093132$$
$$x_{41} = -102.176760093132$$
$$x_{31} = -100.176760093132$$
$$x_{37} = -98.176760093132$$
$$x_{10} = -96.176760093132$$
$$x_{29} = -94.176760093132$$
$$x_{48} = -92.176760093132$$
$$x_{3} = -90.176760093132$$
$$x_{43} = -88.176760093132$$
$$x_{35} = -86.176760093132$$
$$x_{33} = -84.176760093132$$
$$x_{39} = -82.1767600931321$$
$$x_{30} = -80.1767600931323$$
$$x_{20} = -78.1767600931332$$
$$x_{6} = -76.1767600931369$$
$$x_{38} = -74.1767600931515$$
$$x_{42} = -72.1767600932097$$
$$x_{18} = -70.1767600934429$$
$$x_{22} = -68.1767600943754$$
$$x_{46} = -66.1767600981057$$
$$x_{7} = -64.1767601130287$$
$$x_{26} = -62.1767601727313$$
$$x_{24} = -60.1767604116072$$
$$x_{19} = -58.1767613675219$$
$$x_{17} = -56.1767651937535$$
$$x_{47} = -54.1767805148382$$
$$x_{36} = -52.1768419014221$$
$$x_{21} = -50.1770881070134$$
$$x_{44} = -48.1780773770746$$
$$x_{49} = -46.1820676051361$$
$$x_{25} = -44.1983252994765$$
$$x_{23} = -42.2668361314626$$
$$x_{9} = -40.6038236940885$$
$$x_{34} = -7.21435991128888$$
$$x_{28} = 0.535347584231804$$
$$x_{2} = 2.99343804847255$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{32}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{32} - \frac{1}{10}$$
=
$$-130.176760093132 + - \frac{1}{10}$$
=
$$-130.276760093132$$
lo sustituimos en la expresión
$$- 2^{x - 3} + \left(17 \cdot 4^{x} + \left(8 \cdot 5^{x} + \left(- 2 \cdot 8^{x} + \left(- 17 \cdot 10^{x} + 2 \cdot 20^{x}\right)\right)\right)\right) \leq 0$$
$$- 2^{-130.276760093132 - 3} + \left(\left(\left(- \frac{2}{8^{130.276760093132}} + \left(- \frac{17}{10^{130.276760093132}} + \frac{2}{20^{130.276760093132}}\right)\right) + \frac{8}{5^{130.276760093132}}\right) + \frac{17}{4^{130.276760093132}}\right) \leq 0$$
-7.58049362857481e-41 <= 0

significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -130.176760093132$$
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       x32      x14      x5      x12      x16      x15      x1      x13      x11      x40      x8      x27      x4      x45      x41      x31      x37      x10      x29      x48      x3      x43      x35      x33      x39      x30      x20      x6      x38      x42      x18      x22      x46      x7      x26      x24      x19      x17      x47      x36      x21      x44      x49      x25      x23      x9      x34      x28      x2

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -130.176760093132$$
$$x \geq -128.176760093132 \wedge x \leq -126.176760093132$$
$$x \geq -124.176760093132 \wedge x \leq -122.176760093132$$
$$x \geq -120.176760093132 \wedge x \leq -118.176760093132$$
$$x \geq -116.176760093132 \wedge x \leq -114.176760093132$$
$$x \geq -112.176760093132 \wedge x \leq -110.176760093132$$
$$x \geq -108.176760093132 \wedge x \leq -106.176760093132$$
$$x \geq -104.176760093132 \wedge x \leq -102.176760093132$$
$$x \geq -100.176760093132 \wedge x \leq -98.176760093132$$
$$x \geq -96.176760093132 \wedge x \leq -94.176760093132$$
$$x \geq -92.176760093132 \wedge x \leq -90.176760093132$$
$$x \geq -88.176760093132 \wedge x \leq -86.176760093132$$
$$x \geq -84.176760093132 \wedge x \leq -82.1767600931321$$
$$x \geq -80.1767600931323 \wedge x \leq -78.1767600931332$$
$$x \geq -76.1767600931369 \wedge x \leq -74.1767600931515$$
$$x \geq -72.1767600932097 \wedge x \leq -70.1767600934429$$
$$x \geq -68.1767600943754 \wedge x \leq -66.1767600981057$$
$$x \geq -64.1767601130287 \wedge x \leq -62.1767601727313$$
$$x \geq -60.1767604116072 \wedge x \leq -58.1767613675219$$
$$x \geq -56.1767651937535 \wedge x \leq -54.1767805148382$$
$$x \geq -52.1768419014221 \wedge x \leq -50.1770881070134$$
$$x \geq -48.1780773770746 \wedge x \leq -46.1820676051361$$
$$x \geq -44.1983252994765 \wedge x \leq -42.2668361314626$$
$$x \geq -40.6038236940885 \wedge x \leq -7.21435991128888$$
$$x \geq 0.535347584231804 \wedge x \leq 2.99343804847255$$
Solución de la desigualdad en el gráfico