Mister Exam

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

A inequation with variable

### The solution

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  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
Detail solution
Given the inequality:
$$- 3^{x + 4} + \left(81 \cdot 5^{x} + \left(2 \cdot 9^{x + 1} + \left(- 18 \cdot 15^{x} + \left(- 27^{x} + 45^{x}\right)\right)\right)\right) \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 3^{x + 4} + \left(81 \cdot 5^{x} + \left(2 \cdot 9^{x + 1} + \left(- 18 \cdot 15^{x} + \left(- 27^{x} + 45^{x}\right)\right)\right)\right) = 0$$
Solve:
$$x_{1} = 2$$
$$x_{2} = -62.9855570615851$$
$$x_{3} = -104.985557061373$$
$$x_{4} = -54.9855570740071$$
$$x_{5} = -34.9859027016135$$
$$x_{6} = -86.9855570613729$$
$$x_{7} = -32.9865175766309$$
$$x_{8} = -58.9855570630103$$
$$x_{9} = -108.985557061373$$
$$x_{10} = -90.9855570613729$$
$$x_{11} = -44.98555915085$$
$$x_{12} = -112.985557061373$$
$$x_{13} = -92.9855570613729$$
$$x_{14} = -40.9855731840259$$
$$x_{15} = -78.9855570613729$$
$$x_{16} = -46.985557813584$$
$$x_{17} = -56.9855570659212$$
$$x_{18} = -118.985557061373$$
$$x_{19} = -48.9855573321688$$
$$x_{20} = -96.9855570613729$$
$$x_{21} = -82.9855570613729$$
$$x_{22} = -80.9855570613729$$
$$x_{23} = -100.985557061373$$
$$x_{24} = -74.9855570613733$$
$$x_{25} = -102.985557061373$$
$$x_{26} = -116.985557061373$$
$$x_{27} = 0$$
$$x_{28} = -52.985557096468$$
$$x_{29} = -50.9855571588594$$
$$x_{30} = -72.9855570613742$$
$$x_{31} = -36.9856814734066$$
$$x_{32} = -68.9855570613828$$
$$x_{33} = -114.985557061373$$
$$x_{34} = -98.9855570613729$$
$$x_{35} = -66.9855570614004$$
$$x_{36} = -38.9856018473541$$
$$x_{37} = -42.9855628654895$$
$$x_{38} = -64.9855570614493$$
$$x_{39} = -60.9855570619623$$
$$x_{40} = -76.985557061373$$
$$x_{41} = -28.9930026468386$$
$$x_{42} = -88.9855570613729$$
$$x_{43} = -84.9855570613729$$
$$x_{44} = -106.985557061373$$
$$x_{45} = -94.9855570613729$$
$$x_{46} = -70.9855570613764$$
$$x_{47} = -30.9882283475395$$
$$x_{48} = -110.985557061373$$
$$x_{1} = 2$$
$$x_{2} = -62.9855570615851$$
$$x_{3} = -104.985557061373$$
$$x_{4} = -54.9855570740071$$
$$x_{5} = -34.9859027016135$$
$$x_{6} = -86.9855570613729$$
$$x_{7} = -32.9865175766309$$
$$x_{8} = -58.9855570630103$$
$$x_{9} = -108.985557061373$$
$$x_{10} = -90.9855570613729$$
$$x_{11} = -44.98555915085$$
$$x_{12} = -112.985557061373$$
$$x_{13} = -92.9855570613729$$
$$x_{14} = -40.9855731840259$$
$$x_{15} = -78.9855570613729$$
$$x_{16} = -46.985557813584$$
$$x_{17} = -56.9855570659212$$
$$x_{18} = -118.985557061373$$
$$x_{19} = -48.9855573321688$$
$$x_{20} = -96.9855570613729$$
$$x_{21} = -82.9855570613729$$
$$x_{22} = -80.9855570613729$$
$$x_{23} = -100.985557061373$$
$$x_{24} = -74.9855570613733$$
$$x_{25} = -102.985557061373$$
$$x_{26} = -116.985557061373$$
$$x_{27} = 0$$
$$x_{28} = -52.985557096468$$
$$x_{29} = -50.9855571588594$$
$$x_{30} = -72.9855570613742$$
$$x_{31} = -36.9856814734066$$
$$x_{32} = -68.9855570613828$$
$$x_{33} = -114.985557061373$$
$$x_{34} = -98.9855570613729$$
$$x_{35} = -66.9855570614004$$
$$x_{36} = -38.9856018473541$$
$$x_{37} = -42.9855628654895$$
$$x_{38} = -64.9855570614493$$
$$x_{39} = -60.9855570619623$$
$$x_{40} = -76.985557061373$$
$$x_{41} = -28.9930026468386$$
$$x_{42} = -88.9855570613729$$
$$x_{43} = -84.9855570613729$$
$$x_{44} = -106.985557061373$$
$$x_{45} = -94.9855570613729$$
$$x_{46} = -70.9855570613764$$
$$x_{47} = -30.9882283475395$$
$$x_{48} = -110.985557061373$$
This roots
$$x_{18} = -118.985557061373$$
$$x_{26} = -116.985557061373$$
$$x_{33} = -114.985557061373$$
$$x_{12} = -112.985557061373$$
$$x_{48} = -110.985557061373$$
$$x_{9} = -108.985557061373$$
$$x_{44} = -106.985557061373$$
$$x_{3} = -104.985557061373$$
$$x_{25} = -102.985557061373$$
$$x_{23} = -100.985557061373$$
$$x_{34} = -98.9855570613729$$
$$x_{20} = -96.9855570613729$$
$$x_{45} = -94.9855570613729$$
$$x_{13} = -92.9855570613729$$
$$x_{10} = -90.9855570613729$$
$$x_{42} = -88.9855570613729$$
$$x_{6} = -86.9855570613729$$
$$x_{43} = -84.9855570613729$$
$$x_{21} = -82.9855570613729$$
$$x_{22} = -80.9855570613729$$
$$x_{15} = -78.9855570613729$$
$$x_{40} = -76.985557061373$$
$$x_{24} = -74.9855570613733$$
$$x_{30} = -72.9855570613742$$
$$x_{46} = -70.9855570613764$$
$$x_{32} = -68.9855570613828$$
$$x_{35} = -66.9855570614004$$
$$x_{38} = -64.9855570614493$$
$$x_{2} = -62.9855570615851$$
$$x_{39} = -60.9855570619623$$
$$x_{8} = -58.9855570630103$$
$$x_{17} = -56.9855570659212$$
$$x_{4} = -54.9855570740071$$
$$x_{28} = -52.985557096468$$
$$x_{29} = -50.9855571588594$$
$$x_{19} = -48.9855573321688$$
$$x_{16} = -46.985557813584$$
$$x_{11} = -44.98555915085$$
$$x_{37} = -42.9855628654895$$
$$x_{14} = -40.9855731840259$$
$$x_{36} = -38.9856018473541$$
$$x_{31} = -36.9856814734066$$
$$x_{5} = -34.9859027016135$$
$$x_{7} = -32.9865175766309$$
$$x_{47} = -30.9882283475395$$
$$x_{41} = -28.9930026468386$$
$$x_{27} = 0$$
$$x_{1} = 2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{18}$$
For example, let's take the point
$$x_{0} = x_{18} - \frac{1}{10}$$
=
$$-118.985557061373 + - \frac{1}{10}$$
=
$$-119.085557061373$$
substitute to the expression
$$- 3^{x + 4} + \left(81 \cdot 5^{x} + \left(2 \cdot 9^{x + 1} + \left(- 18 \cdot 15^{x} + \left(- 27^{x} + 45^{x}\right)\right)\right)\right) \leq 0$$
$$- 3^{-119.085557061373 + 4} + \left(\left(\left(- \frac{18}{15^{119.085557061373}} + \left(- \frac{1}{27^{119.085557061373}} + 45^{-119.085557061373}\right)\right) + 2 \cdot 9^{-119.085557061373 + 1}\right) + \frac{81}{5^{119.085557061373}}\right) \leq 0$$
-1.23093356673045e-55 <= 0

one of the solutions of our inequality is:
$$x \leq -118.985557061373$$
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x18      x26      x33      x12      x48      x9      x44      x3      x25      x23      x34      x20      x45      x13      x10      x42      x6      x43      x21      x22      x15      x40      x24      x30      x46      x32      x35      x38      x2      x39      x8      x17      x4      x28      x29      x19      x16      x11      x37      x14      x36      x31      x5      x7      x47      x41      x27      x1

Other solutions will get with the changeover to the next point
etc.
$$x \leq -118.985557061373$$
$$x \geq -116.985557061373 \wedge x \leq -114.985557061373$$
$$x \geq -112.985557061373 \wedge x \leq -110.985557061373$$
$$x \geq -108.985557061373 \wedge x \leq -106.985557061373$$
$$x \geq -104.985557061373 \wedge x \leq -102.985557061373$$
$$x \geq -100.985557061373 \wedge x \leq -98.9855570613729$$
$$x \geq -96.9855570613729 \wedge x \leq -94.9855570613729$$
$$x \geq -92.9855570613729 \wedge x \leq -90.9855570613729$$
$$x \geq -88.9855570613729 \wedge x \leq -86.9855570613729$$
$$x \geq -84.9855570613729 \wedge x \leq -82.9855570613729$$
$$x \geq -80.9855570613729 \wedge x \leq -78.9855570613729$$
$$x \geq -76.985557061373 \wedge x \leq -74.9855570613733$$
$$x \geq -72.9855570613742 \wedge x \leq -70.9855570613764$$
$$x \geq -68.9855570613828 \wedge x \leq -66.9855570614004$$
$$x \geq -64.9855570614493 \wedge x \leq -62.9855570615851$$
$$x \geq -60.9855570619623 \wedge x \leq -58.9855570630103$$
$$x \geq -56.9855570659212 \wedge x \leq -54.9855570740071$$
$$x \geq -52.985557096468 \wedge x \leq -50.9855571588594$$
$$x \geq -48.9855573321688 \wedge x \leq -46.985557813584$$
$$x \geq -44.98555915085 \wedge x \leq -42.9855628654895$$
$$x \geq -40.9855731840259 \wedge x \leq -38.9856018473541$$
$$x \geq -36.9856814734066 \wedge x \leq -34.9859027016135$$
$$x \geq -32.9865175766309 \wedge x \leq -30.9882283475395$$
$$x \geq -28.9930026468386 \wedge x \leq 0$$
$$x \geq 2$$
Solving inequality on a graph
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