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45^x-27^x-18*15^x+2*9^(x+1)+81*5^x-3^(x+4)<=0

45^x-27^x-18*15^x+2*9^(x+1)+81*5^x-3^(x+4)<=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
  x     x        x      x + 1       x    x + 4     
45  - 27  - 18*15  + 2*9      + 81*5  - 3      <= 0
$$- 3^{x + 4} + \left(81 \cdot 5^{x} + \left(2 \cdot 9^{x + 1} + \left(- 18 \cdot 15^{x} + \left(- 27^{x} + 45^{x}\right)\right)\right)\right) \leq 0$$
-3^(x + 4) + 81*5^x + 2*9^(x + 1) - 18*15^x - 27^x + 45^x <= 0
Solución detallada
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:
$$x \leq -118.985557061373$$
$$x \geq -116.985557061373 \wedge x \leq -114.985557061373$$
$$x \geq -112.985557061373 \wedge x \leq -110.985557061373$$
$$x \geq -108.985557061373 \wedge x \leq -106.985557061373$$
$$x \geq -104.985557061373 \wedge x \leq -102.985557061373$$
$$x \geq -100.985557061373 \wedge x \leq -98.9855570613729$$
$$x \geq -96.9855570613729 \wedge x \leq -94.9855570613729$$
$$x \geq -92.9855570613729 \wedge x \leq -90.9855570613729$$
$$x \geq -88.9855570613729 \wedge x \leq -86.9855570613729$$
$$x \geq -84.9855570613729 \wedge x \leq -82.9855570613729$$
$$x \geq -80.9855570613729 \wedge x \leq -78.9855570613729$$
$$x \geq -76.985557061373 \wedge x \leq -74.9855570613733$$
$$x \geq -72.9855570613742 \wedge x \leq -70.9855570613764$$
$$x \geq -68.9855570613828 \wedge x \leq -66.9855570614004$$
$$x \geq -64.9855570614493 \wedge x \leq -62.9855570615851$$
$$x \geq -60.9855570619623 \wedge x \leq -58.9855570630103$$
$$x \geq -56.9855570659212 \wedge x \leq -54.9855570740071$$
$$x \geq -52.985557096468 \wedge x \leq -50.9855571588594$$
$$x \geq -48.9855573321688 \wedge x \leq -46.985557813584$$
$$x \geq -44.98555915085 \wedge x \leq -42.9855628654895$$
$$x \geq -40.9855731840259 \wedge x \leq -38.9856018473541$$
$$x \geq -36.9856814734066 \wedge x \leq -34.9859027016135$$
$$x \geq -32.9865175766309 \wedge x \leq -30.9882283475395$$
$$x \geq -28.9930026468386 \wedge x \leq 0$$
$$x \geq 2$$
Solución de la desigualdad en el gráfico
Gráfico
45^x-27^x-18*15^x+2*9^(x+1)+81*5^x-3^(x+4)<=0 desigualdades