Se da la desigualdad:
$$- 3^{x + 4} + \left(81 \cdot 5^{x} + \left(2 \cdot 9^{x + 1} + \left(- 18 \cdot 15^{x} + \left(- 27^{x} + 45^{x}\right)\right)\right)\right) \leq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$- 3^{x + 4} + \left(81 \cdot 5^{x} + \left(2 \cdot 9^{x + 1} + \left(- 18 \cdot 15^{x} + \left(- 27^{x} + 45^{x}\right)\right)\right)\right) = 0$$
Resolvemos:
$$x_{1} = -82.9855570613729$$
$$x_{2} = -64.9855570614493$$
$$x_{3} = -80.9855570613729$$
$$x_{4} = -116.985557061373$$
$$x_{5} = -62.9855570615851$$
$$x_{6} = -36.9856814734066$$
$$x_{7} = -34.9859027016135$$
$$x_{8} = -58.9855570630103$$
$$x_{9} = -84.9855570613729$$
$$x_{10} = -42.9855628654895$$
$$x_{11} = -112.985557061373$$
$$x_{12} = -118.985557061373$$
$$x_{13} = -78.9855570613729$$
$$x_{14} = -28.9930026468386$$
$$x_{15} = -40.9855731840259$$
$$x_{16} = -88.9855570613729$$
$$x_{17} = -76.985557061373$$
$$x_{18} = -92.9855570613729$$
$$x_{19} = -104.985557061373$$
$$x_{20} = -30.9882283475395$$
$$x_{21} = -98.9855570613729$$
$$x_{22} = -108.985557061373$$
$$x_{23} = -60.9855570619623$$
$$x_{24} = -94.9855570613729$$
$$x_{25} = -44.98555915085$$
$$x_{26} = -32.9865175766309$$
$$x_{27} = -66.9855570614004$$
$$x_{28} = -48.9855573321688$$
$$x_{29} = 0$$
$$x_{30} = -46.985557813584$$
$$x_{31} = -74.9855570613733$$
$$x_{32} = -100.985557061373$$
$$x_{33} = -114.985557061373$$
$$x_{34} = -50.9855571588594$$
$$x_{35} = -106.985557061373$$
$$x_{36} = -102.985557061373$$
$$x_{37} = -110.985557061373$$
$$x_{38} = -56.9855570659212$$
$$x_{39} = -52.985557096468$$
$$x_{40} = -90.9855570613729$$
$$x_{41} = -70.9855570613764$$
$$x_{42} = -68.9855570613828$$
$$x_{43} = -96.9855570613729$$
$$x_{44} = 2$$
$$x_{45} = -38.9856018473541$$
$$x_{46} = -72.9855570613742$$
$$x_{47} = -54.9855570740071$$
$$x_{48} = -86.9855570613729$$
$$x_{1} = -82.9855570613729$$
$$x_{2} = -64.9855570614493$$
$$x_{3} = -80.9855570613729$$
$$x_{4} = -116.985557061373$$
$$x_{5} = -62.9855570615851$$
$$x_{6} = -36.9856814734066$$
$$x_{7} = -34.9859027016135$$
$$x_{8} = -58.9855570630103$$
$$x_{9} = -84.9855570613729$$
$$x_{10} = -42.9855628654895$$
$$x_{11} = -112.985557061373$$
$$x_{12} = -118.985557061373$$
$$x_{13} = -78.9855570613729$$
$$x_{14} = -28.9930026468386$$
$$x_{15} = -40.9855731840259$$
$$x_{16} = -88.9855570613729$$
$$x_{17} = -76.985557061373$$
$$x_{18} = -92.9855570613729$$
$$x_{19} = -104.985557061373$$
$$x_{20} = -30.9882283475395$$
$$x_{21} = -98.9855570613729$$
$$x_{22} = -108.985557061373$$
$$x_{23} = -60.9855570619623$$
$$x_{24} = -94.9855570613729$$
$$x_{25} = -44.98555915085$$
$$x_{26} = -32.9865175766309$$
$$x_{27} = -66.9855570614004$$
$$x_{28} = -48.9855573321688$$
$$x_{29} = 0$$
$$x_{30} = -46.985557813584$$
$$x_{31} = -74.9855570613733$$
$$x_{32} = -100.985557061373$$
$$x_{33} = -114.985557061373$$
$$x_{34} = -50.9855571588594$$
$$x_{35} = -106.985557061373$$
$$x_{36} = -102.985557061373$$
$$x_{37} = -110.985557061373$$
$$x_{38} = -56.9855570659212$$
$$x_{39} = -52.985557096468$$
$$x_{40} = -90.9855570613729$$
$$x_{41} = -70.9855570613764$$
$$x_{42} = -68.9855570613828$$
$$x_{43} = -96.9855570613729$$
$$x_{44} = 2$$
$$x_{45} = -38.9856018473541$$
$$x_{46} = -72.9855570613742$$
$$x_{47} = -54.9855570740071$$
$$x_{48} = -86.9855570613729$$
Las raíces dadas
$$x_{12} = -118.985557061373$$
$$x_{4} = -116.985557061373$$
$$x_{33} = -114.985557061373$$
$$x_{11} = -112.985557061373$$
$$x_{37} = -110.985557061373$$
$$x_{22} = -108.985557061373$$
$$x_{35} = -106.985557061373$$
$$x_{19} = -104.985557061373$$
$$x_{36} = -102.985557061373$$
$$x_{32} = -100.985557061373$$
$$x_{21} = -98.9855570613729$$
$$x_{43} = -96.9855570613729$$
$$x_{24} = -94.9855570613729$$
$$x_{18} = -92.9855570613729$$
$$x_{40} = -90.9855570613729$$
$$x_{16} = -88.9855570613729$$
$$x_{48} = -86.9855570613729$$
$$x_{9} = -84.9855570613729$$
$$x_{1} = -82.9855570613729$$
$$x_{3} = -80.9855570613729$$
$$x_{13} = -78.9855570613729$$
$$x_{17} = -76.985557061373$$
$$x_{31} = -74.9855570613733$$
$$x_{46} = -72.9855570613742$$
$$x_{41} = -70.9855570613764$$
$$x_{42} = -68.9855570613828$$
$$x_{27} = -66.9855570614004$$
$$x_{2} = -64.9855570614493$$
$$x_{5} = -62.9855570615851$$
$$x_{23} = -60.9855570619623$$
$$x_{8} = -58.9855570630103$$
$$x_{38} = -56.9855570659212$$
$$x_{47} = -54.9855570740071$$
$$x_{39} = -52.985557096468$$
$$x_{34} = -50.9855571588594$$
$$x_{28} = -48.9855573321688$$
$$x_{30} = -46.985557813584$$
$$x_{25} = -44.98555915085$$
$$x_{10} = -42.9855628654895$$
$$x_{15} = -40.9855731840259$$
$$x_{45} = -38.9856018473541$$
$$x_{6} = -36.9856814734066$$
$$x_{7} = -34.9859027016135$$
$$x_{26} = -32.9865175766309$$
$$x_{20} = -30.9882283475395$$
$$x_{14} = -28.9930026468386$$
$$x_{29} = 0$$
$$x_{44} = 2$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{12}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{12} - \frac{1}{10}$$
=
$$-118.985557061373 + - \frac{1}{10}$$
=
$$-119.085557061373$$
lo sustituimos en la expresión
$$- 3^{x + 4} + \left(81 \cdot 5^{x} + \left(2 \cdot 9^{x + 1} + \left(- 18 \cdot 15^{x} + \left(- 27^{x} + 45^{x}\right)\right)\right)\right) \leq 0$$
$$- 3^{-119.085557061373 + 4} + \left(\left(\left(- \frac{18}{15^{119.085557061373}} + \left(- \frac{1}{27^{119.085557061373}} + 45^{-119.085557061373}\right)\right) + 2 \cdot 9^{-119.085557061373 + 1}\right) + \frac{81}{5^{119.085557061373}}\right) \leq 0$$
-1.23093356673045e-55 <= 0
significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -118.985557061373$$
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x12 x4 x33 x11 x37 x22 x35 x19 x36 x32 x21 x43 x24 x18 x40 x16 x48 x9 x1 x3 x13 x17 x31 x46 x41 x42 x27 x2 x5 x23 x8 x38 x47 x39 x34 x28 x30 x25 x10 x15 x45 x6 x7 x26 x20 x14 x29 x44
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -118.985557061373$$
$$x \geq -116.985557061373 \wedge x \leq -114.985557061373$$
$$x \geq -112.985557061373 \wedge x \leq -110.985557061373$$
$$x \geq -108.985557061373 \wedge x \leq -106.985557061373$$
$$x \geq -104.985557061373 \wedge x \leq -102.985557061373$$
$$x \geq -100.985557061373 \wedge x \leq -98.9855570613729$$
$$x \geq -96.9855570613729 \wedge x \leq -94.9855570613729$$
$$x \geq -92.9855570613729 \wedge x \leq -90.9855570613729$$
$$x \geq -88.9855570613729 \wedge x \leq -86.9855570613729$$
$$x \geq -84.9855570613729 \wedge x \leq -82.9855570613729$$
$$x \geq -80.9855570613729 \wedge x \leq -78.9855570613729$$
$$x \geq -76.985557061373 \wedge x \leq -74.9855570613733$$
$$x \geq -72.9855570613742 \wedge x \leq -70.9855570613764$$
$$x \geq -68.9855570613828 \wedge x \leq -66.9855570614004$$
$$x \geq -64.9855570614493 \wedge x \leq -62.9855570615851$$
$$x \geq -60.9855570619623 \wedge x \leq -58.9855570630103$$
$$x \geq -56.9855570659212 \wedge x \leq -54.9855570740071$$
$$x \geq -52.985557096468 \wedge x \leq -50.9855571588594$$
$$x \geq -48.9855573321688 \wedge x \leq -46.985557813584$$
$$x \geq -44.98555915085 \wedge x \leq -42.9855628654895$$
$$x \geq -40.9855731840259 \wedge x \leq -38.9856018473541$$
$$x \geq -36.9856814734066 \wedge x \leq -34.9859027016135$$
$$x \geq -32.9865175766309 \wedge x \leq -30.9882283475395$$
$$x \geq -28.9930026468386 \wedge x \leq 0$$
$$x \geq 2$$