Mister Exam
(-3^(4*x-x^2-3)-1)*log(1/2)(x^2-4*x+5)>=0 inequation
The solution
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/ 2 \ | 4*x - x - 3 | / 2 \ \- 3 - 1/*log(1/2)*\x - 4*x + 5/ >= 0
$$\left(- 3^{\left(- x^{2} + 4 x\right) - 3} - 1\right) \log{\left(\frac{1}{2} \right)} \left(\left(x^{2} - 4 x\right) + 5\right) \geq 0$$
((-3^(-x^2 + 4*x - 3) - 1)*log(1/2))*(x^2 - 4*x + 5) >= 0
Detail solution
Given the inequality:
$$\left(- 3^{\left(- x^{2} + 4 x\right) - 3} - 1\right) \log{\left(\frac{1}{2} \right)} \left(\left(x^{2} - 4 x\right) + 5\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 3^{\left(- x^{2} + 4 x\right) - 3} - 1\right) \log{\left(\frac{1}{2} \right)} \left(\left(x^{2} - 4 x\right) + 5\right) = 0$$
Solve:
$$x_{1} = 2 - i$$
$$x_{2} = 2 + i$$
$$x_{3} = 2 - \frac{\sqrt{\log{\left(3 \right)} - i \pi}}{\sqrt{\log{\left(3 \right)}}}$$
$$x_{4} = 2 + \frac{\sqrt{\log{\left(3 \right)} - i \pi}}{\sqrt{\log{\left(3 \right)}}}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
$$\left(-1 - 3^{-3 + \left(0 \cdot 4 - 0^{2}\right)}\right) \log{\left(\frac{1}{2} \right)} \left(\left(0^{2} - 0 \cdot 4\right) + 5\right) \geq 0$$
so the inequality is always executed
$$\left(- 3^{\left(- x^{2} + 4 x\right) - 3} - 1\right) \log{\left(\frac{1}{2} \right)} \left(\left(x^{2} - 4 x\right) + 5\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 3^{\left(- x^{2} + 4 x\right) - 3} - 1\right) \log{\left(\frac{1}{2} \right)} \left(\left(x^{2} - 4 x\right) + 5\right) = 0$$
Solve:
$$x_{1} = 2 - i$$
$$x_{2} = 2 + i$$
$$x_{3} = 2 - \frac{\sqrt{\log{\left(3 \right)} - i \pi}}{\sqrt{\log{\left(3 \right)}}}$$
$$x_{4} = 2 + \frac{\sqrt{\log{\left(3 \right)} - i \pi}}{\sqrt{\log{\left(3 \right)}}}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
x0 = 0
$$\left(-1 - 3^{-3 + \left(0 \cdot 4 - 0^{2}\right)}\right) \log{\left(\frac{1}{2} \right)} \left(\left(0^{2} - 0 \cdot 4\right) + 5\right) \geq 0$$
140*log(2)
---------- >= 0
27 so the inequality is always executed
Solving inequality on a graph