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3*45^x-3*27^x-28*15^x+28*9^x+9*5^x-3^(x+2)<=0

3*45^x-3*27^x-28*15^x+28*9^x+9*5^x-3^(x+2)<=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
    x       x        x       x      x    x + 2     
3*45  - 3*27  - 28*15  + 28*9  + 9*5  - 3      <= 0
$$- 3^{x + 2} + \left(9 \cdot 5^{x} + \left(28 \cdot 9^{x} + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) \leq 0$$
-3^(x + 2) + 9*5^x + 28*9^x - 28*15^x - 3*27^x + 3*45^x <= 0
Solución detallada
Se da la desigualdad:
$$- 3^{x + 2} + \left(9 \cdot 5^{x} + \left(28 \cdot 9^{x} + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) \leq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$- 3^{x + 2} + \left(9 \cdot 5^{x} + \left(28 \cdot 9^{x} + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) = 0$$
Resolvemos:
$$x_{1} = -104.985557061373$$
$$x_{2} = -52.985557096468$$
$$x_{3} = -68.9855570613828$$
$$x_{4} = -96.9855570613729$$
$$x_{5} = -94.9855570613729$$
$$x_{6} = -100.985557061373$$
$$x_{7} = -56.9855570659212$$
$$x_{8} = -98.9855570613729$$
$$x_{9} = -32.9367772221347$$
$$x_{10} = -90.9855570613729$$
$$x_{11} = -32.9865185933132$$
$$x_{12} = -66.9855570614004$$
$$x_{13} = -34.9859028145641$$
$$x_{14} = -36.9856814859561$$
$$x_{15} = -118.985557061373$$
$$x_{16} = -116.985557061373$$
$$x_{17} = -70.9855570613764$$
$$x_{18} = -38.9856018487484$$
$$x_{19} = -76.985557061373$$
$$x_{20} = -50.9855571588594$$
$$x_{21} = -46.9855578135842$$
$$x_{22} = -40.9855731841808$$
$$x_{23} = -30.98823750091$$
$$x_{24} = -78.9855570613729$$
$$x_{25} = -82.9855570613729$$
$$x_{26} = -80.9855570613729$$
$$x_{27} = -42.9855628655068$$
$$x_{28} = -74.9855570613733$$
$$x_{29} = -112.985557061373$$
$$x_{30} = -86.9855570613729$$
$$x_{31} = -110.985557061373$$
$$x_{32} = -84.9855570613729$$
$$x_{33} = 2$$
$$x_{34} = -64.9855570614493$$
$$x_{35} = -92.9855570613729$$
$$x_{36} = -108.985557061373$$
$$x_{37} = -88.9855570613729$$
$$x_{38} = -72.9855570613742$$
$$x_{39} = -48.9855573321688$$
$$x_{40} = -106.985557061373$$
$$x_{41} = -62.9855570615851$$
$$x_{42} = -27.0071877297463$$
$$x_{43} = -114.985557061373$$
$$x_{44} = -60.9855570619623$$
$$x_{45} = -44.9855591508519$$
$$x_{46} = -58.9855570630103$$
$$x_{47} = 0$$
$$x_{48} = -54.9855570740071$$
$$x_{49} = -102.985557061373$$
$$x_{50} = -28.9930851119168$$
$$x_{1} = -104.985557061373$$
$$x_{2} = -52.985557096468$$
$$x_{3} = -68.9855570613828$$
$$x_{4} = -96.9855570613729$$
$$x_{5} = -94.9855570613729$$
$$x_{6} = -100.985557061373$$
$$x_{7} = -56.9855570659212$$
$$x_{8} = -98.9855570613729$$
$$x_{9} = -32.9367772221347$$
$$x_{10} = -90.9855570613729$$
$$x_{11} = -32.9865185933132$$
$$x_{12} = -66.9855570614004$$
$$x_{13} = -34.9859028145641$$
$$x_{14} = -36.9856814859561$$
$$x_{15} = -118.985557061373$$
$$x_{16} = -116.985557061373$$
$$x_{17} = -70.9855570613764$$
$$x_{18} = -38.9856018487484$$
$$x_{19} = -76.985557061373$$
$$x_{20} = -50.9855571588594$$
$$x_{21} = -46.9855578135842$$
$$x_{22} = -40.9855731841808$$
$$x_{23} = -30.98823750091$$
$$x_{24} = -78.9855570613729$$
$$x_{25} = -82.9855570613729$$
$$x_{26} = -80.9855570613729$$
$$x_{27} = -42.9855628655068$$
$$x_{28} = -74.9855570613733$$
$$x_{29} = -112.985557061373$$
$$x_{30} = -86.9855570613729$$
$$x_{31} = -110.985557061373$$
$$x_{32} = -84.9855570613729$$
$$x_{33} = 2$$
$$x_{34} = -64.9855570614493$$
$$x_{35} = -92.9855570613729$$
$$x_{36} = -108.985557061373$$
$$x_{37} = -88.9855570613729$$
$$x_{38} = -72.9855570613742$$
$$x_{39} = -48.9855573321688$$
$$x_{40} = -106.985557061373$$
$$x_{41} = -62.9855570615851$$
$$x_{42} = -27.0071877297463$$
$$x_{43} = -114.985557061373$$
$$x_{44} = -60.9855570619623$$
$$x_{45} = -44.9855591508519$$
$$x_{46} = -58.9855570630103$$
$$x_{47} = 0$$
$$x_{48} = -54.9855570740071$$
$$x_{49} = -102.985557061373$$
$$x_{50} = -28.9930851119168$$
Las raíces dadas
$$x_{15} = -118.985557061373$$
$$x_{16} = -116.985557061373$$
$$x_{43} = -114.985557061373$$
$$x_{29} = -112.985557061373$$
$$x_{31} = -110.985557061373$$
$$x_{36} = -108.985557061373$$
$$x_{40} = -106.985557061373$$
$$x_{1} = -104.985557061373$$
$$x_{49} = -102.985557061373$$
$$x_{6} = -100.985557061373$$
$$x_{8} = -98.9855570613729$$
$$x_{4} = -96.9855570613729$$
$$x_{5} = -94.9855570613729$$
$$x_{35} = -92.9855570613729$$
$$x_{10} = -90.9855570613729$$
$$x_{37} = -88.9855570613729$$
$$x_{30} = -86.9855570613729$$
$$x_{32} = -84.9855570613729$$
$$x_{25} = -82.9855570613729$$
$$x_{26} = -80.9855570613729$$
$$x_{24} = -78.9855570613729$$
$$x_{19} = -76.985557061373$$
$$x_{28} = -74.9855570613733$$
$$x_{38} = -72.9855570613742$$
$$x_{17} = -70.9855570613764$$
$$x_{3} = -68.9855570613828$$
$$x_{12} = -66.9855570614004$$
$$x_{34} = -64.9855570614493$$
$$x_{41} = -62.9855570615851$$
$$x_{44} = -60.9855570619623$$
$$x_{46} = -58.9855570630103$$
$$x_{7} = -56.9855570659212$$
$$x_{48} = -54.9855570740071$$
$$x_{2} = -52.985557096468$$
$$x_{20} = -50.9855571588594$$
$$x_{39} = -48.9855573321688$$
$$x_{21} = -46.9855578135842$$
$$x_{45} = -44.9855591508519$$
$$x_{27} = -42.9855628655068$$
$$x_{22} = -40.9855731841808$$
$$x_{18} = -38.9856018487484$$
$$x_{14} = -36.9856814859561$$
$$x_{13} = -34.9859028145641$$
$$x_{11} = -32.9865185933132$$
$$x_{9} = -32.9367772221347$$
$$x_{23} = -30.98823750091$$
$$x_{50} = -28.9930851119168$$
$$x_{42} = -27.0071877297463$$
$$x_{47} = 0$$
$$x_{33} = 2$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{15}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{15} - \frac{1}{10}$$
=
$$-118.985557061373 + - \frac{1}{10}$$
=
$$-119.085557061373$$
lo sustituimos en la expresión
$$- 3^{x + 2} + \left(9 \cdot 5^{x} + \left(28 \cdot 9^{x} + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) \leq 0$$
$$- 3^{-119.085557061373 + 2} + \left(\left(\left(- \frac{28}{15^{119.085557061373}} + \left(- \frac{3}{27^{119.085557061373}} + \frac{3}{45^{119.085557061373}}\right)\right) + \frac{28}{9^{119.085557061373}}\right) + \frac{9}{5^{119.085557061373}}\right) \leq 0$$
-1.36770396303383e-56 <= 0

significa que una de las soluciones de nuestra ecuación será con:
$$x \leq -118.985557061373$$
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       x15      x16      x43      x29      x31      x36      x40      x1      x49      x6      x8      x4      x5      x35      x10      x37      x30      x32      x25      x26      x24      x19      x28      x38      x17      x3      x12      x34      x41      x44      x46      x7      x48      x2      x20      x39      x21      x45      x27      x22      x18      x14      x13      x11      x9      x23      x50      x42      x47      x33

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \leq -118.985557061373$$
$$x \geq -116.985557061373 \wedge x \leq -114.985557061373$$
$$x \geq -112.985557061373 \wedge x \leq -110.985557061373$$
$$x \geq -108.985557061373 \wedge x \leq -106.985557061373$$
$$x \geq -104.985557061373 \wedge x \leq -102.985557061373$$
$$x \geq -100.985557061373 \wedge x \leq -98.9855570613729$$
$$x \geq -96.9855570613729 \wedge x \leq -94.9855570613729$$
$$x \geq -92.9855570613729 \wedge x \leq -90.9855570613729$$
$$x \geq -88.9855570613729 \wedge x \leq -86.9855570613729$$
$$x \geq -84.9855570613729 \wedge x \leq -82.9855570613729$$
$$x \geq -80.9855570613729 \wedge x \leq -78.9855570613729$$
$$x \geq -76.985557061373 \wedge x \leq -74.9855570613733$$
$$x \geq -72.9855570613742 \wedge x \leq -70.9855570613764$$
$$x \geq -68.9855570613828 \wedge x \leq -66.9855570614004$$
$$x \geq -64.9855570614493 \wedge x \leq -62.9855570615851$$
$$x \geq -60.9855570619623 \wedge x \leq -58.9855570630103$$
$$x \geq -56.9855570659212 \wedge x \leq -54.9855570740071$$
$$x \geq -52.985557096468 \wedge x \leq -50.9855571588594$$
$$x \geq -48.9855573321688 \wedge x \leq -46.9855578135842$$
$$x \geq -44.9855591508519 \wedge x \leq -42.9855628655068$$
$$x \geq -40.9855731841808 \wedge x \leq -38.9856018487484$$
$$x \geq -36.9856814859561 \wedge x \leq -34.9859028145641$$
$$x \geq -32.9865185933132 \wedge x \leq -32.9367772221347$$
$$x \geq -30.98823750091 \wedge x \leq -28.9930851119168$$
$$x \geq -27.0071877297463 \wedge x \leq 0$$
$$x \geq 2$$
Solución de la desigualdad en el gráfico
Gráfico
3*45^x-3*27^x-28*15^x+28*9^x+9*5^x-3^(x+2)<=0 desigualdades