Mister Exam
3*45^x-3*27^x-28*15^x+28*9*x+9*5^x-3^x*9<0 inequation
The solution
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x x x x x 3*45 - 3*27 - 28*15 + 252*x + 9*5 - 3 *9 < 0
$$- 9 \cdot 3^{x} + \left(9 \cdot 5^{x} + \left(252 x + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) < 0$$
-9*3^x + 9*5^x + 252*x - 28*15^x - 3*27^x + 3*45^x < 0
Detail solution
Given the inequality:
$$- 9 \cdot 3^{x} + \left(9 \cdot 5^{x} + \left(252 x + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 9 \cdot 3^{x} + \left(9 \cdot 5^{x} + \left(252 x + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) = 0$$
Solve:
$$x_{1} = 0.170485635999203$$
$$x_{2} = 2.31304520170066$$
$$x_{1} = 0.170485635999203$$
$$x_{2} = 2.31304520170066$$
This roots
$$x_{1} = 0.170485635999203$$
$$x_{2} = 2.31304520170066$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.170485635999203$$
=
$$0.0704856359992029$$
substitute to the expression
$$- 9 \cdot 3^{x} + \left(9 \cdot 5^{x} + \left(252 x + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) < 0$$
$$- 9 \cdot 3^{0.0704856359992029} + \left(\left(\left(- 28 \cdot 15^{0.0704856359992029} + \left(- 3 \cdot 27^{0.0704856359992029} + 3 \cdot 45^{0.0704856359992029}\right)\right) + 0.0704856359992029 \cdot 252\right) + 9 \cdot 5^{0.0704856359992029}\right) < 0$$
one of the solutions of our inequality is:
$$x < 0.170485635999203$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 0.170485635999203$$
$$x > 2.31304520170066$$
$$- 9 \cdot 3^{x} + \left(9 \cdot 5^{x} + \left(252 x + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 9 \cdot 3^{x} + \left(9 \cdot 5^{x} + \left(252 x + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) = 0$$
Solve:
$$x_{1} = 0.170485635999203$$
$$x_{2} = 2.31304520170066$$
$$x_{1} = 0.170485635999203$$
$$x_{2} = 2.31304520170066$$
This roots
$$x_{1} = 0.170485635999203$$
$$x_{2} = 2.31304520170066$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.170485635999203$$
=
$$0.0704856359992029$$
substitute to the expression
$$- 9 \cdot 3^{x} + \left(9 \cdot 5^{x} + \left(252 x + \left(- 28 \cdot 15^{x} + \left(- 3 \cdot 27^{x} + 3 \cdot 45^{x}\right)\right)\right)\right) < 0$$
$$- 9 \cdot 3^{0.0704856359992029} + \left(\left(\left(- 28 \cdot 15^{0.0704856359992029} + \left(- 3 \cdot 27^{0.0704856359992029} + 3 \cdot 45^{0.0704856359992029}\right)\right) + 0.0704856359992029 \cdot 252\right) + 9 \cdot 5^{0.0704856359992029}\right) < 0$$
-15.6310998859040 < 0
one of the solutions of our inequality is:
$$x < 0.170485635999203$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 0.170485635999203$$
$$x > 2.31304520170066$$