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:
$$x \leq -130.176760093132$$
$$x \geq -128.176760093132 \wedge x \leq -126.176760093132$$
$$x \geq -124.176760093132 \wedge x \leq -122.176760093132$$
$$x \geq -120.176760093132 \wedge x \leq -118.176760093132$$
$$x \geq -116.176760093132 \wedge x \leq -114.176760093132$$
$$x \geq -112.176760093132 \wedge x \leq -110.176760093132$$
$$x \geq -108.176760093132 \wedge x \leq -106.176760093132$$
$$x \geq -104.176760093132 \wedge x \leq -102.176760093132$$
$$x \geq -100.176760093132 \wedge x \leq -98.176760093132$$
$$x \geq -96.176760093132 \wedge x \leq -94.176760093132$$
$$x \geq -92.176760093132 \wedge x \leq -90.176760093132$$
$$x \geq -88.176760093132 \wedge x \leq -86.176760093132$$
$$x \geq -84.176760093132 \wedge x \leq -82.1767600931321$$
$$x \geq -80.1767600931323 \wedge x \leq -78.1767600931332$$
$$x \geq -76.1767600931369 \wedge x \leq -74.1767600931515$$
$$x \geq -72.1767600932097 \wedge x \leq -70.1767600934429$$
$$x \geq -68.1767600943754 \wedge x \leq -66.1767600981057$$
$$x \geq -64.1767601130287 \wedge x \leq -62.1767601727313$$
$$x \geq -60.1767604116072 \wedge x \leq -58.1767613675219$$
$$x \geq -56.1767651937535 \wedge x \leq -54.1767805148382$$
$$x \geq -52.1768419014221 \wedge x \leq -50.1770881070134$$
$$x \geq -48.1780773770746 \wedge x \leq -46.1820676051361$$
$$x \geq -44.1983252994765 \wedge x \leq -42.2668361314626$$
$$x \geq -40.6038236940885 \wedge x \leq -7.21435991128888$$
$$x \geq 0.535347584231804 \wedge x \leq 2.99343804847255$$