Mister Exam
log(0,6)(x-1)/(3^x-4)*(|x|-2)<0 inequation
The solution
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log(3/5)*(x - 1)
----------------*(|x| - 2) < 0
x
3 - 4
$$\frac{\left(x - 1\right) \log{\left(\frac{3}{5} \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) < 0$$
(((x - 1)*log(3/5))/(3^x - 4))*(|x| - 2) < 0
Detail solution
Given the inequality:
$$\frac{\left(x - 1\right) \log{\left(\frac{3}{5} \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x - 1\right) \log{\left(\frac{3}{5} \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) = 0$$
Solve:
$$x_{1} = 117.63065008493$$
$$x_{2} = 54.1754041498667$$
$$x_{3} = 81.8093351823563$$
$$x_{4} = 83.7945417792735$$
$$x_{5} = 1$$
$$x_{6} = 91.7430230340759$$
$$x_{7} = 44.4790720428965$$
$$x_{8} = 71.8988127461799$$
$$x_{9} = 87.7673771375313$$
$$x_{10} = 46.4023616016304$$
$$x_{11} = 113.644114134408$$
$$x_{12} = 56.1326398893666$$
$$x_{13} = 58.0938754502403$$
$$x_{14} = 75.8595124437987$$
$$x_{15} = 105.674574148931$$
$$x_{16} = 107.666470345888$$
$$x_{17} = 67.9440310092733$$
$$x_{18} = 85.780578489646$$
$$x_{19} = 35.120171048217$$
$$x_{20} = 93.7317667140894$$
$$x_{21} = 31.6501211792559$$
$$x_{22} = 109.658706085884$$
$$x_{23} = 115.637249469329$$
$$x_{24} = 119.624300873587$$
$$x_{25} = 77.8417284674232$$
$$x_{26} = 52.2228296756877$$
$$x_{27} = 73.8784982850986$$
$$x_{28} = 38.792646877732$$
$$x_{29} = 33.3488814376598$$
$$x_{30} = 65.9692963096669$$
$$x_{31} = 63.9966231594372$$
$$x_{32} = 48.3351441758331$$
$$x_{33} = 111.65126043137$$
$$x_{34} = 2$$
$$x_{35} = 42.5674973423199$$
$$x_{36} = 101.691893748395$$
$$x_{37} = 89.7548768356836$$
$$x_{38} = 79.8250352924933$$
$$x_{39} = 50.2757346429048$$
$$x_{40} = 103.683040312127$$
$$x_{41} = 69.9206011629382$$
$$x_{42} = 97.7108740878187$$
$$x_{43} = 60.0585699759989$$
$$x_{44} = 62.0262766940749$$
$$x_{45} = -2$$
$$x_{46} = 99.7011617133151$$
$$x_{47} = 95.7210637029333$$
$$x_{48} = 36.9395050123055$$
$$x_{49} = 40.6706356213694$$
$$x_{1} = 117.63065008493$$
$$x_{2} = 54.1754041498667$$
$$x_{3} = 81.8093351823563$$
$$x_{4} = 83.7945417792735$$
$$x_{5} = 1$$
$$x_{6} = 91.7430230340759$$
$$x_{7} = 44.4790720428965$$
$$x_{8} = 71.8988127461799$$
$$x_{9} = 87.7673771375313$$
$$x_{10} = 46.4023616016304$$
$$x_{11} = 113.644114134408$$
$$x_{12} = 56.1326398893666$$
$$x_{13} = 58.0938754502403$$
$$x_{14} = 75.8595124437987$$
$$x_{15} = 105.674574148931$$
$$x_{16} = 107.666470345888$$
$$x_{17} = 67.9440310092733$$
$$x_{18} = 85.780578489646$$
$$x_{19} = 35.120171048217$$
$$x_{20} = 93.7317667140894$$
$$x_{21} = 31.6501211792559$$
$$x_{22} = 109.658706085884$$
$$x_{23} = 115.637249469329$$
$$x_{24} = 119.624300873587$$
$$x_{25} = 77.8417284674232$$
$$x_{26} = 52.2228296756877$$
$$x_{27} = 73.8784982850986$$
$$x_{28} = 38.792646877732$$
$$x_{29} = 33.3488814376598$$
$$x_{30} = 65.9692963096669$$
$$x_{31} = 63.9966231594372$$
$$x_{32} = 48.3351441758331$$
$$x_{33} = 111.65126043137$$
$$x_{34} = 2$$
$$x_{35} = 42.5674973423199$$
$$x_{36} = 101.691893748395$$
$$x_{37} = 89.7548768356836$$
$$x_{38} = 79.8250352924933$$
$$x_{39} = 50.2757346429048$$
$$x_{40} = 103.683040312127$$
$$x_{41} = 69.9206011629382$$
$$x_{42} = 97.7108740878187$$
$$x_{43} = 60.0585699759989$$
$$x_{44} = 62.0262766940749$$
$$x_{45} = -2$$
$$x_{46} = 99.7011617133151$$
$$x_{47} = 95.7210637029333$$
$$x_{48} = 36.9395050123055$$
$$x_{49} = 40.6706356213694$$
This roots
$$x_{45} = -2$$
$$x_{5} = 1$$
$$x_{34} = 2$$
$$x_{21} = 31.6501211792559$$
$$x_{29} = 33.3488814376598$$
$$x_{19} = 35.120171048217$$
$$x_{48} = 36.9395050123055$$
$$x_{28} = 38.792646877732$$
$$x_{49} = 40.6706356213694$$
$$x_{35} = 42.5674973423199$$
$$x_{7} = 44.4790720428965$$
$$x_{10} = 46.4023616016304$$
$$x_{32} = 48.3351441758331$$
$$x_{39} = 50.2757346429048$$
$$x_{26} = 52.2228296756877$$
$$x_{2} = 54.1754041498667$$
$$x_{12} = 56.1326398893666$$
$$x_{13} = 58.0938754502403$$
$$x_{43} = 60.0585699759989$$
$$x_{44} = 62.0262766940749$$
$$x_{31} = 63.9966231594372$$
$$x_{30} = 65.9692963096669$$
$$x_{17} = 67.9440310092733$$
$$x_{41} = 69.9206011629382$$
$$x_{8} = 71.8988127461799$$
$$x_{27} = 73.8784982850986$$
$$x_{14} = 75.8595124437987$$
$$x_{25} = 77.8417284674232$$
$$x_{38} = 79.8250352924933$$
$$x_{3} = 81.8093351823563$$
$$x_{4} = 83.7945417792735$$
$$x_{18} = 85.780578489646$$
$$x_{9} = 87.7673771375313$$
$$x_{37} = 89.7548768356836$$
$$x_{6} = 91.7430230340759$$
$$x_{20} = 93.7317667140894$$
$$x_{47} = 95.7210637029333$$
$$x_{42} = 97.7108740878187$$
$$x_{46} = 99.7011617133151$$
$$x_{36} = 101.691893748395$$
$$x_{40} = 103.683040312127$$
$$x_{15} = 105.674574148931$$
$$x_{16} = 107.666470345888$$
$$x_{22} = 109.658706085884$$
$$x_{33} = 111.65126043137$$
$$x_{11} = 113.644114134408$$
$$x_{23} = 115.637249469329$$
$$x_{1} = 117.63065008493$$
$$x_{24} = 119.624300873587$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{45}$$
For example, let's take the point
$$x_{0} = x_{45} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$-2.1$$
substitute to the expression
$$\frac{\left(x - 1\right) \log{\left(\frac{3}{5} \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) < 0$$
$$\frac{\left(-2.1 - 1\right) \log{\left(\frac{3}{5} \right)}}{-4 + 3^{-2.1}} \left(-2 + \left|{-2.1}\right|\right) < 0$$
one of the solutions of our inequality is:
$$x < -2$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -2$$
$$x > 1 \wedge x < 2$$
$$x > 31.6501211792559 \wedge x < 33.3488814376598$$
$$x > 35.120171048217 \wedge x < 36.9395050123055$$
$$x > 38.792646877732 \wedge x < 40.6706356213694$$
$$x > 42.5674973423199 \wedge x < 44.4790720428965$$
$$x > 46.4023616016304 \wedge x < 48.3351441758331$$
$$x > 50.2757346429048 \wedge x < 52.2228296756877$$
$$x > 54.1754041498667 \wedge x < 56.1326398893666$$
$$x > 58.0938754502403 \wedge x < 60.0585699759989$$
$$x > 62.0262766940749 \wedge x < 63.9966231594372$$
$$x > 65.9692963096669 \wedge x < 67.9440310092733$$
$$x > 69.9206011629382 \wedge x < 71.8988127461799$$
$$x > 73.8784982850986 \wedge x < 75.8595124437987$$
$$x > 77.8417284674232 \wedge x < 79.8250352924933$$
$$x > 81.8093351823563 \wedge x < 83.7945417792735$$
$$x > 85.780578489646 \wedge x < 87.7673771375313$$
$$x > 89.7548768356836 \wedge x < 91.7430230340759$$
$$x > 93.7317667140894 \wedge x < 95.7210637029333$$
$$x > 97.7108740878187 \wedge x < 99.7011617133151$$
$$x > 101.691893748395 \wedge x < 103.683040312127$$
$$x > 105.674574148931 \wedge x < 107.666470345888$$
$$x > 109.658706085884 \wedge x < 111.65126043137$$
$$x > 113.644114134408 \wedge x < 115.637249469329$$
$$x > 117.63065008493 \wedge x < 119.624300873587$$
$$\frac{\left(x - 1\right) \log{\left(\frac{3}{5} \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x - 1\right) \log{\left(\frac{3}{5} \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) = 0$$
Solve:
$$x_{1} = 117.63065008493$$
$$x_{2} = 54.1754041498667$$
$$x_{3} = 81.8093351823563$$
$$x_{4} = 83.7945417792735$$
$$x_{5} = 1$$
$$x_{6} = 91.7430230340759$$
$$x_{7} = 44.4790720428965$$
$$x_{8} = 71.8988127461799$$
$$x_{9} = 87.7673771375313$$
$$x_{10} = 46.4023616016304$$
$$x_{11} = 113.644114134408$$
$$x_{12} = 56.1326398893666$$
$$x_{13} = 58.0938754502403$$
$$x_{14} = 75.8595124437987$$
$$x_{15} = 105.674574148931$$
$$x_{16} = 107.666470345888$$
$$x_{17} = 67.9440310092733$$
$$x_{18} = 85.780578489646$$
$$x_{19} = 35.120171048217$$
$$x_{20} = 93.7317667140894$$
$$x_{21} = 31.6501211792559$$
$$x_{22} = 109.658706085884$$
$$x_{23} = 115.637249469329$$
$$x_{24} = 119.624300873587$$
$$x_{25} = 77.8417284674232$$
$$x_{26} = 52.2228296756877$$
$$x_{27} = 73.8784982850986$$
$$x_{28} = 38.792646877732$$
$$x_{29} = 33.3488814376598$$
$$x_{30} = 65.9692963096669$$
$$x_{31} = 63.9966231594372$$
$$x_{32} = 48.3351441758331$$
$$x_{33} = 111.65126043137$$
$$x_{34} = 2$$
$$x_{35} = 42.5674973423199$$
$$x_{36} = 101.691893748395$$
$$x_{37} = 89.7548768356836$$
$$x_{38} = 79.8250352924933$$
$$x_{39} = 50.2757346429048$$
$$x_{40} = 103.683040312127$$
$$x_{41} = 69.9206011629382$$
$$x_{42} = 97.7108740878187$$
$$x_{43} = 60.0585699759989$$
$$x_{44} = 62.0262766940749$$
$$x_{45} = -2$$
$$x_{46} = 99.7011617133151$$
$$x_{47} = 95.7210637029333$$
$$x_{48} = 36.9395050123055$$
$$x_{49} = 40.6706356213694$$
$$x_{1} = 117.63065008493$$
$$x_{2} = 54.1754041498667$$
$$x_{3} = 81.8093351823563$$
$$x_{4} = 83.7945417792735$$
$$x_{5} = 1$$
$$x_{6} = 91.7430230340759$$
$$x_{7} = 44.4790720428965$$
$$x_{8} = 71.8988127461799$$
$$x_{9} = 87.7673771375313$$
$$x_{10} = 46.4023616016304$$
$$x_{11} = 113.644114134408$$
$$x_{12} = 56.1326398893666$$
$$x_{13} = 58.0938754502403$$
$$x_{14} = 75.8595124437987$$
$$x_{15} = 105.674574148931$$
$$x_{16} = 107.666470345888$$
$$x_{17} = 67.9440310092733$$
$$x_{18} = 85.780578489646$$
$$x_{19} = 35.120171048217$$
$$x_{20} = 93.7317667140894$$
$$x_{21} = 31.6501211792559$$
$$x_{22} = 109.658706085884$$
$$x_{23} = 115.637249469329$$
$$x_{24} = 119.624300873587$$
$$x_{25} = 77.8417284674232$$
$$x_{26} = 52.2228296756877$$
$$x_{27} = 73.8784982850986$$
$$x_{28} = 38.792646877732$$
$$x_{29} = 33.3488814376598$$
$$x_{30} = 65.9692963096669$$
$$x_{31} = 63.9966231594372$$
$$x_{32} = 48.3351441758331$$
$$x_{33} = 111.65126043137$$
$$x_{34} = 2$$
$$x_{35} = 42.5674973423199$$
$$x_{36} = 101.691893748395$$
$$x_{37} = 89.7548768356836$$
$$x_{38} = 79.8250352924933$$
$$x_{39} = 50.2757346429048$$
$$x_{40} = 103.683040312127$$
$$x_{41} = 69.9206011629382$$
$$x_{42} = 97.7108740878187$$
$$x_{43} = 60.0585699759989$$
$$x_{44} = 62.0262766940749$$
$$x_{45} = -2$$
$$x_{46} = 99.7011617133151$$
$$x_{47} = 95.7210637029333$$
$$x_{48} = 36.9395050123055$$
$$x_{49} = 40.6706356213694$$
This roots
$$x_{45} = -2$$
$$x_{5} = 1$$
$$x_{34} = 2$$
$$x_{21} = 31.6501211792559$$
$$x_{29} = 33.3488814376598$$
$$x_{19} = 35.120171048217$$
$$x_{48} = 36.9395050123055$$
$$x_{28} = 38.792646877732$$
$$x_{49} = 40.6706356213694$$
$$x_{35} = 42.5674973423199$$
$$x_{7} = 44.4790720428965$$
$$x_{10} = 46.4023616016304$$
$$x_{32} = 48.3351441758331$$
$$x_{39} = 50.2757346429048$$
$$x_{26} = 52.2228296756877$$
$$x_{2} = 54.1754041498667$$
$$x_{12} = 56.1326398893666$$
$$x_{13} = 58.0938754502403$$
$$x_{43} = 60.0585699759989$$
$$x_{44} = 62.0262766940749$$
$$x_{31} = 63.9966231594372$$
$$x_{30} = 65.9692963096669$$
$$x_{17} = 67.9440310092733$$
$$x_{41} = 69.9206011629382$$
$$x_{8} = 71.8988127461799$$
$$x_{27} = 73.8784982850986$$
$$x_{14} = 75.8595124437987$$
$$x_{25} = 77.8417284674232$$
$$x_{38} = 79.8250352924933$$
$$x_{3} = 81.8093351823563$$
$$x_{4} = 83.7945417792735$$
$$x_{18} = 85.780578489646$$
$$x_{9} = 87.7673771375313$$
$$x_{37} = 89.7548768356836$$
$$x_{6} = 91.7430230340759$$
$$x_{20} = 93.7317667140894$$
$$x_{47} = 95.7210637029333$$
$$x_{42} = 97.7108740878187$$
$$x_{46} = 99.7011617133151$$
$$x_{36} = 101.691893748395$$
$$x_{40} = 103.683040312127$$
$$x_{15} = 105.674574148931$$
$$x_{16} = 107.666470345888$$
$$x_{22} = 109.658706085884$$
$$x_{33} = 111.65126043137$$
$$x_{11} = 113.644114134408$$
$$x_{23} = 115.637249469329$$
$$x_{1} = 117.63065008493$$
$$x_{24} = 119.624300873587$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{45}$$
For example, let's take the point
$$x_{0} = x_{45} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$-2.1$$
substitute to the expression
$$\frac{\left(x - 1\right) \log{\left(\frac{3}{5} \right)}}{3^{x} - 4} \left(\left|{x}\right| - 2\right) < 0$$
$$\frac{\left(-2.1 - 1\right) \log{\left(\frac{3}{5} \right)}}{-4 + 3^{-2.1}} \left(-2 + \left|{-2.1}\right|\right) < 0$$
0.0794780281013577*log(3/5) < 0
one of the solutions of our inequality is:
$$x < -2$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------ο-------
x45 x5 x34 x21 x29 x19 x48 x28 x49 x35 x7 x10 x32 x39 x26 x2 x12 x13 x43 x44 x31 x30 x17 x41 x8 x27 x14 x25 x38 x3 x4 x18 x9 x37 x6 x20 x47 x42 x46 x36 x40 x15 x16 x22 x33 x11 x23 x1 x24Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -2$$
$$x > 1 \wedge x < 2$$
$$x > 31.6501211792559 \wedge x < 33.3488814376598$$
$$x > 35.120171048217 \wedge x < 36.9395050123055$$
$$x > 38.792646877732 \wedge x < 40.6706356213694$$
$$x > 42.5674973423199 \wedge x < 44.4790720428965$$
$$x > 46.4023616016304 \wedge x < 48.3351441758331$$
$$x > 50.2757346429048 \wedge x < 52.2228296756877$$
$$x > 54.1754041498667 \wedge x < 56.1326398893666$$
$$x > 58.0938754502403 \wedge x < 60.0585699759989$$
$$x > 62.0262766940749 \wedge x < 63.9966231594372$$
$$x > 65.9692963096669 \wedge x < 67.9440310092733$$
$$x > 69.9206011629382 \wedge x < 71.8988127461799$$
$$x > 73.8784982850986 \wedge x < 75.8595124437987$$
$$x > 77.8417284674232 \wedge x < 79.8250352924933$$
$$x > 81.8093351823563 \wedge x < 83.7945417792735$$
$$x > 85.780578489646 \wedge x < 87.7673771375313$$
$$x > 89.7548768356836 \wedge x < 91.7430230340759$$
$$x > 93.7317667140894 \wedge x < 95.7210637029333$$
$$x > 97.7108740878187 \wedge x < 99.7011617133151$$
$$x > 101.691893748395 \wedge x < 103.683040312127$$
$$x > 105.674574148931 \wedge x < 107.666470345888$$
$$x > 109.658706085884 \wedge x < 111.65126043137$$
$$x > 113.644114134408 \wedge x < 115.637249469329$$
$$x > 117.63065008493 \wedge x < 119.624300873587$$
Solving inequality on a graph
Rapid solution 2
[src]
log(4)
(-oo, -2) U (1, ------) U (2, oo)
log(3)
$$x\ in\ \left(-\infty, -2\right) \cup \left(1, \frac{\log{\left(4 \right)}}{\log{\left(3 \right)}}\right) \cup \left(2, \infty\right)$$
x in Union(Interval.open(-oo, -2), Interval.open(1, log(4)/log(3)), Interval.open(2, oo))
Rapid solution
[src]
/ / log(4)\ \ Or|And|1 < x, x < ------|, And(2 < x, x < oo), x < -2| \ \ log(3)/ /
$$\left(1 < x \wedge x < \frac{\log{\left(4 \right)}}{\log{\left(3 \right)}}\right) \vee \left(2 < x \wedge x < \infty\right) \vee x < -2$$
(x < -2)∨((2 < x)∧(x < oo))∨((1 < x)∧(x < log(4)/log(3)))