Mister Exam
6•4^x+6^x-2•9^x>=0 inequation
The solution
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x x x 6*4 + 6 - 2*9 >= 0
$$- 2 \cdot 9^{x} + \left(6 \cdot 4^{x} + 6^{x}\right) \geq 0$$
-2*9^x + 6*4^x + 6^x >= 0
Detail solution
Given the inequality:
$$- 2 \cdot 9^{x} + \left(6 \cdot 4^{x} + 6^{x}\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \cdot 9^{x} + \left(6 \cdot 4^{x} + 6^{x}\right) = 0$$
Solve:
$$x_{1} = -103.016148295613$$
$$x_{2} = -45.0161475312093$$
$$x_{3} = -39.0161395897739$$
$$x_{4} = -29.0156505597641$$
$$x_{5} = -97.0161482956131$$
$$x_{6} = -47.0161479558756$$
$$x_{7} = -23.0109910424438$$
$$x_{8} = -71.0161482955929$$
$$x_{9} = -35.0161042493011$$
$$x_{10} = -55.016148282357$$
$$x_{11} = -57.0161482897215$$
$$x_{12} = -75.0161482956091$$
$$x_{13} = -61.0161482944493$$
$$x_{14} = -49.0161481446181$$
$$x_{15} = -65.0161482953832$$
$$x_{16} = -113.016148295613$$
$$x_{17} = -33.0160492860239$$
$$x_{18} = -95.0161482956131$$
$$x_{19} = -101.016148295613$$
$$x_{20} = -87.016148295613$$
$$x_{21} = -109.016148295613$$
$$x_{22} = -69.0161482955677$$
$$x_{23} = -25.0137179041395$$
$$x_{24} = -83.0161482956129$$
$$x_{25} = -41.0161444260271$$
$$x_{26} = -63.0161482950958$$
$$x_{27} = -91.0161482956131$$
$$x_{28} = -93.0161482956131$$
$$x_{29} = -51.0161482285041$$
$$x_{30} = -89.0161482956131$$
$$x_{31} = -105.016148295613$$
$$x_{32} = -73.0161482956041$$
$$x_{33} = -27.0150406393294$$
$$x_{34} = -107.016148295613$$
$$x_{35} = -43.016146575733$$
$$x_{36} = -111.016148295613$$
$$x_{37} = -31.01592600289$$
$$x_{38} = -99.0161482956131$$
$$x_{39} = -53.0161482657869$$
$$x_{40} = -115.016148295613$$
$$x_{41} = -81.0161482956127$$
$$x_{42} = -85.016148295613$$
$$x_{43} = -37.0161287111672$$
$$x_{44} = -67.0161482955109$$
$$x_{45} = -79.0161482956123$$
$$x_{46} = -77.0161482956113$$
$$x_{47} = -59.0161482929946$$
$$x_{48} = 1.70951129135145$$
$$x_{1} = -103.016148295613$$
$$x_{2} = -45.0161475312093$$
$$x_{3} = -39.0161395897739$$
$$x_{4} = -29.0156505597641$$
$$x_{5} = -97.0161482956131$$
$$x_{6} = -47.0161479558756$$
$$x_{7} = -23.0109910424438$$
$$x_{8} = -71.0161482955929$$
$$x_{9} = -35.0161042493011$$
$$x_{10} = -55.016148282357$$
$$x_{11} = -57.0161482897215$$
$$x_{12} = -75.0161482956091$$
$$x_{13} = -61.0161482944493$$
$$x_{14} = -49.0161481446181$$
$$x_{15} = -65.0161482953832$$
$$x_{16} = -113.016148295613$$
$$x_{17} = -33.0160492860239$$
$$x_{18} = -95.0161482956131$$
$$x_{19} = -101.016148295613$$
$$x_{20} = -87.016148295613$$
$$x_{21} = -109.016148295613$$
$$x_{22} = -69.0161482955677$$
$$x_{23} = -25.0137179041395$$
$$x_{24} = -83.0161482956129$$
$$x_{25} = -41.0161444260271$$
$$x_{26} = -63.0161482950958$$
$$x_{27} = -91.0161482956131$$
$$x_{28} = -93.0161482956131$$
$$x_{29} = -51.0161482285041$$
$$x_{30} = -89.0161482956131$$
$$x_{31} = -105.016148295613$$
$$x_{32} = -73.0161482956041$$
$$x_{33} = -27.0150406393294$$
$$x_{34} = -107.016148295613$$
$$x_{35} = -43.016146575733$$
$$x_{36} = -111.016148295613$$
$$x_{37} = -31.01592600289$$
$$x_{38} = -99.0161482956131$$
$$x_{39} = -53.0161482657869$$
$$x_{40} = -115.016148295613$$
$$x_{41} = -81.0161482956127$$
$$x_{42} = -85.016148295613$$
$$x_{43} = -37.0161287111672$$
$$x_{44} = -67.0161482955109$$
$$x_{45} = -79.0161482956123$$
$$x_{46} = -77.0161482956113$$
$$x_{47} = -59.0161482929946$$
$$x_{48} = 1.70951129135145$$
This roots
$$x_{40} = -115.016148295613$$
$$x_{16} = -113.016148295613$$
$$x_{36} = -111.016148295613$$
$$x_{21} = -109.016148295613$$
$$x_{34} = -107.016148295613$$
$$x_{31} = -105.016148295613$$
$$x_{1} = -103.016148295613$$
$$x_{19} = -101.016148295613$$
$$x_{38} = -99.0161482956131$$
$$x_{5} = -97.0161482956131$$
$$x_{18} = -95.0161482956131$$
$$x_{28} = -93.0161482956131$$
$$x_{27} = -91.0161482956131$$
$$x_{30} = -89.0161482956131$$
$$x_{20} = -87.016148295613$$
$$x_{42} = -85.016148295613$$
$$x_{24} = -83.0161482956129$$
$$x_{41} = -81.0161482956127$$
$$x_{45} = -79.0161482956123$$
$$x_{46} = -77.0161482956113$$
$$x_{12} = -75.0161482956091$$
$$x_{32} = -73.0161482956041$$
$$x_{8} = -71.0161482955929$$
$$x_{22} = -69.0161482955677$$
$$x_{44} = -67.0161482955109$$
$$x_{15} = -65.0161482953832$$
$$x_{26} = -63.0161482950958$$
$$x_{13} = -61.0161482944493$$
$$x_{47} = -59.0161482929946$$
$$x_{11} = -57.0161482897215$$
$$x_{10} = -55.016148282357$$
$$x_{39} = -53.0161482657869$$
$$x_{29} = -51.0161482285041$$
$$x_{14} = -49.0161481446181$$
$$x_{6} = -47.0161479558756$$
$$x_{2} = -45.0161475312093$$
$$x_{35} = -43.016146575733$$
$$x_{25} = -41.0161444260271$$
$$x_{3} = -39.0161395897739$$
$$x_{43} = -37.0161287111672$$
$$x_{9} = -35.0161042493011$$
$$x_{17} = -33.0160492860239$$
$$x_{37} = -31.01592600289$$
$$x_{4} = -29.0156505597641$$
$$x_{33} = -27.0150406393294$$
$$x_{23} = -25.0137179041395$$
$$x_{7} = -23.0109910424438$$
$$x_{48} = 1.70951129135145$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{40}$$
For example, let's take the point
$$x_{0} = x_{40} - \frac{1}{10}$$
=
$$-115.016148295613 + - \frac{1}{10}$$
=
$$-115.116148295613$$
substitute to the expression
$$- 2 \cdot 9^{x} + \left(6 \cdot 4^{x} + 6^{x}\right) \geq 0$$
$$- \frac{2}{9^{115.116148295613}} + \left(6^{-115.116148295613} + \frac{6}{4^{115.116148295613}}\right) \geq 0$$
one of the solutions of our inequality is:
$$x \leq -115.016148295613$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -115.016148295613$$
$$x \geq -113.016148295613 \wedge x \leq -111.016148295613$$
$$x \geq -109.016148295613 \wedge x \leq -107.016148295613$$
$$x \geq -105.016148295613 \wedge x \leq -103.016148295613$$
$$x \geq -101.016148295613 \wedge x \leq -99.0161482956131$$
$$x \geq -97.0161482956131 \wedge x \leq -95.0161482956131$$
$$x \geq -93.0161482956131 \wedge x \leq -91.0161482956131$$
$$x \geq -89.0161482956131 \wedge x \leq -87.016148295613$$
$$x \geq -85.016148295613 \wedge x \leq -83.0161482956129$$
$$x \geq -81.0161482956127 \wedge x \leq -79.0161482956123$$
$$x \geq -77.0161482956113 \wedge x \leq -75.0161482956091$$
$$x \geq -73.0161482956041 \wedge x \leq -71.0161482955929$$
$$x \geq -69.0161482955677 \wedge x \leq -67.0161482955109$$
$$x \geq -65.0161482953832 \wedge x \leq -63.0161482950958$$
$$x \geq -61.0161482944493 \wedge x \leq -59.0161482929946$$
$$x \geq -57.0161482897215 \wedge x \leq -55.016148282357$$
$$x \geq -53.0161482657869 \wedge x \leq -51.0161482285041$$
$$x \geq -49.0161481446181 \wedge x \leq -47.0161479558756$$
$$x \geq -45.0161475312093 \wedge x \leq -43.016146575733$$
$$x \geq -41.0161444260271 \wedge x \leq -39.0161395897739$$
$$x \geq -37.0161287111672 \wedge x \leq -35.0161042493011$$
$$x \geq -33.0160492860239 \wedge x \leq -31.01592600289$$
$$x \geq -29.0156505597641 \wedge x \leq -27.0150406393294$$
$$x \geq -25.0137179041395 \wedge x \leq -23.0109910424438$$
$$x \geq 1.70951129135145$$
$$- 2 \cdot 9^{x} + \left(6 \cdot 4^{x} + 6^{x}\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 2 \cdot 9^{x} + \left(6 \cdot 4^{x} + 6^{x}\right) = 0$$
Solve:
$$x_{1} = -103.016148295613$$
$$x_{2} = -45.0161475312093$$
$$x_{3} = -39.0161395897739$$
$$x_{4} = -29.0156505597641$$
$$x_{5} = -97.0161482956131$$
$$x_{6} = -47.0161479558756$$
$$x_{7} = -23.0109910424438$$
$$x_{8} = -71.0161482955929$$
$$x_{9} = -35.0161042493011$$
$$x_{10} = -55.016148282357$$
$$x_{11} = -57.0161482897215$$
$$x_{12} = -75.0161482956091$$
$$x_{13} = -61.0161482944493$$
$$x_{14} = -49.0161481446181$$
$$x_{15} = -65.0161482953832$$
$$x_{16} = -113.016148295613$$
$$x_{17} = -33.0160492860239$$
$$x_{18} = -95.0161482956131$$
$$x_{19} = -101.016148295613$$
$$x_{20} = -87.016148295613$$
$$x_{21} = -109.016148295613$$
$$x_{22} = -69.0161482955677$$
$$x_{23} = -25.0137179041395$$
$$x_{24} = -83.0161482956129$$
$$x_{25} = -41.0161444260271$$
$$x_{26} = -63.0161482950958$$
$$x_{27} = -91.0161482956131$$
$$x_{28} = -93.0161482956131$$
$$x_{29} = -51.0161482285041$$
$$x_{30} = -89.0161482956131$$
$$x_{31} = -105.016148295613$$
$$x_{32} = -73.0161482956041$$
$$x_{33} = -27.0150406393294$$
$$x_{34} = -107.016148295613$$
$$x_{35} = -43.016146575733$$
$$x_{36} = -111.016148295613$$
$$x_{37} = -31.01592600289$$
$$x_{38} = -99.0161482956131$$
$$x_{39} = -53.0161482657869$$
$$x_{40} = -115.016148295613$$
$$x_{41} = -81.0161482956127$$
$$x_{42} = -85.016148295613$$
$$x_{43} = -37.0161287111672$$
$$x_{44} = -67.0161482955109$$
$$x_{45} = -79.0161482956123$$
$$x_{46} = -77.0161482956113$$
$$x_{47} = -59.0161482929946$$
$$x_{48} = 1.70951129135145$$
$$x_{1} = -103.016148295613$$
$$x_{2} = -45.0161475312093$$
$$x_{3} = -39.0161395897739$$
$$x_{4} = -29.0156505597641$$
$$x_{5} = -97.0161482956131$$
$$x_{6} = -47.0161479558756$$
$$x_{7} = -23.0109910424438$$
$$x_{8} = -71.0161482955929$$
$$x_{9} = -35.0161042493011$$
$$x_{10} = -55.016148282357$$
$$x_{11} = -57.0161482897215$$
$$x_{12} = -75.0161482956091$$
$$x_{13} = -61.0161482944493$$
$$x_{14} = -49.0161481446181$$
$$x_{15} = -65.0161482953832$$
$$x_{16} = -113.016148295613$$
$$x_{17} = -33.0160492860239$$
$$x_{18} = -95.0161482956131$$
$$x_{19} = -101.016148295613$$
$$x_{20} = -87.016148295613$$
$$x_{21} = -109.016148295613$$
$$x_{22} = -69.0161482955677$$
$$x_{23} = -25.0137179041395$$
$$x_{24} = -83.0161482956129$$
$$x_{25} = -41.0161444260271$$
$$x_{26} = -63.0161482950958$$
$$x_{27} = -91.0161482956131$$
$$x_{28} = -93.0161482956131$$
$$x_{29} = -51.0161482285041$$
$$x_{30} = -89.0161482956131$$
$$x_{31} = -105.016148295613$$
$$x_{32} = -73.0161482956041$$
$$x_{33} = -27.0150406393294$$
$$x_{34} = -107.016148295613$$
$$x_{35} = -43.016146575733$$
$$x_{36} = -111.016148295613$$
$$x_{37} = -31.01592600289$$
$$x_{38} = -99.0161482956131$$
$$x_{39} = -53.0161482657869$$
$$x_{40} = -115.016148295613$$
$$x_{41} = -81.0161482956127$$
$$x_{42} = -85.016148295613$$
$$x_{43} = -37.0161287111672$$
$$x_{44} = -67.0161482955109$$
$$x_{45} = -79.0161482956123$$
$$x_{46} = -77.0161482956113$$
$$x_{47} = -59.0161482929946$$
$$x_{48} = 1.70951129135145$$
This roots
$$x_{40} = -115.016148295613$$
$$x_{16} = -113.016148295613$$
$$x_{36} = -111.016148295613$$
$$x_{21} = -109.016148295613$$
$$x_{34} = -107.016148295613$$
$$x_{31} = -105.016148295613$$
$$x_{1} = -103.016148295613$$
$$x_{19} = -101.016148295613$$
$$x_{38} = -99.0161482956131$$
$$x_{5} = -97.0161482956131$$
$$x_{18} = -95.0161482956131$$
$$x_{28} = -93.0161482956131$$
$$x_{27} = -91.0161482956131$$
$$x_{30} = -89.0161482956131$$
$$x_{20} = -87.016148295613$$
$$x_{42} = -85.016148295613$$
$$x_{24} = -83.0161482956129$$
$$x_{41} = -81.0161482956127$$
$$x_{45} = -79.0161482956123$$
$$x_{46} = -77.0161482956113$$
$$x_{12} = -75.0161482956091$$
$$x_{32} = -73.0161482956041$$
$$x_{8} = -71.0161482955929$$
$$x_{22} = -69.0161482955677$$
$$x_{44} = -67.0161482955109$$
$$x_{15} = -65.0161482953832$$
$$x_{26} = -63.0161482950958$$
$$x_{13} = -61.0161482944493$$
$$x_{47} = -59.0161482929946$$
$$x_{11} = -57.0161482897215$$
$$x_{10} = -55.016148282357$$
$$x_{39} = -53.0161482657869$$
$$x_{29} = -51.0161482285041$$
$$x_{14} = -49.0161481446181$$
$$x_{6} = -47.0161479558756$$
$$x_{2} = -45.0161475312093$$
$$x_{35} = -43.016146575733$$
$$x_{25} = -41.0161444260271$$
$$x_{3} = -39.0161395897739$$
$$x_{43} = -37.0161287111672$$
$$x_{9} = -35.0161042493011$$
$$x_{17} = -33.0160492860239$$
$$x_{37} = -31.01592600289$$
$$x_{4} = -29.0156505597641$$
$$x_{33} = -27.0150406393294$$
$$x_{23} = -25.0137179041395$$
$$x_{7} = -23.0109910424438$$
$$x_{48} = 1.70951129135145$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{40}$$
For example, let's take the point
$$x_{0} = x_{40} - \frac{1}{10}$$
=
$$-115.016148295613 + - \frac{1}{10}$$
=
$$-115.116148295613$$
substitute to the expression
$$- 2 \cdot 9^{x} + \left(6 \cdot 4^{x} + 6^{x}\right) \geq 0$$
$$- \frac{2}{9^{115.116148295613}} + \left(6^{-115.116148295613} + \frac{6}{4^{115.116148295613}}\right) \geq 0$$
2.96022011802196e-69 >= 0
one of the solutions of our inequality is:
$$x \leq -115.016148295613$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
x40 x16 x36 x21 x34 x31 x1 x19 x38 x5 x18 x28 x27 x30 x20 x42 x24 x41 x45 x46 x12 x32 x8 x22 x44 x15 x26 x13 x47 x11 x10 x39 x29 x14 x6 x2 x35 x25 x3 x43 x9 x17 x37 x4 x33 x23 x7 x48Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -115.016148295613$$
$$x \geq -113.016148295613 \wedge x \leq -111.016148295613$$
$$x \geq -109.016148295613 \wedge x \leq -107.016148295613$$
$$x \geq -105.016148295613 \wedge x \leq -103.016148295613$$
$$x \geq -101.016148295613 \wedge x \leq -99.0161482956131$$
$$x \geq -97.0161482956131 \wedge x \leq -95.0161482956131$$
$$x \geq -93.0161482956131 \wedge x \leq -91.0161482956131$$
$$x \geq -89.0161482956131 \wedge x \leq -87.016148295613$$
$$x \geq -85.016148295613 \wedge x \leq -83.0161482956129$$
$$x \geq -81.0161482956127 \wedge x \leq -79.0161482956123$$
$$x \geq -77.0161482956113 \wedge x \leq -75.0161482956091$$
$$x \geq -73.0161482956041 \wedge x \leq -71.0161482955929$$
$$x \geq -69.0161482955677 \wedge x \leq -67.0161482955109$$
$$x \geq -65.0161482953832 \wedge x \leq -63.0161482950958$$
$$x \geq -61.0161482944493 \wedge x \leq -59.0161482929946$$
$$x \geq -57.0161482897215 \wedge x \leq -55.016148282357$$
$$x \geq -53.0161482657869 \wedge x \leq -51.0161482285041$$
$$x \geq -49.0161481446181 \wedge x \leq -47.0161479558756$$
$$x \geq -45.0161475312093 \wedge x \leq -43.016146575733$$
$$x \geq -41.0161444260271 \wedge x \leq -39.0161395897739$$
$$x \geq -37.0161287111672 \wedge x \leq -35.0161042493011$$
$$x \geq -33.0160492860239 \wedge x \leq -31.01592600289$$
$$x \geq -29.0156505597641 \wedge x \leq -27.0150406393294$$
$$x \geq -25.0137179041395 \wedge x \leq -23.0109910424438$$
$$x \geq 1.70951129135145$$
Solving inequality on a graph