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

one of the solutions of our inequality is:
$$x \leq -118.985557061373$$
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x_30      x_14      x_39      x_15      x_44      x_18      x_43      x_45      x_10      x_37      x_42      x_24      x_13      x_27      x_9      x_5      x_12      x_11      x_20      x_48      x_4      x_38      x_33      x_26      x_1      x_16      x_17      x_19      x_41      x_23      x_22      x_21      x_29      x_6      x_32      x_7      x_3      x_47      x_34      x_46      x_36      x_31      x_35      x_40      x_8      x_28      x_2      x_25

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
The graph