Mister Exam
(2+sqrt(3))^x+3(2-sqrt(3))^x<0 inequation
The solution
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x x / ___\ / ___\ \2 + \/ 3 / + 3*\2 - \/ 3 / < 0
$$3 \left(2 - \sqrt{3}\right)^{x} + \left(\sqrt{3} + 2\right)^{x} < 0$$
3*(2 - sqrt(3))^x + (sqrt(3) + 2)^x < 0
Detail solution
Given the inequality:
$$3 \left(2 - \sqrt{3}\right)^{x} + \left(\sqrt{3} + 2\right)^{x} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$3 \left(2 - \sqrt{3}\right)^{x} + \left(\sqrt{3} + 2\right)^{x} = 0$$
Solve:
$$x_{1} = \frac{\log{\left(3 \right)} + i \pi}{\log{\left(4 \sqrt{3} + 7 \right)}}$$
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
$$\left(\sqrt{3} + 2\right)^{0} + 3 \left(2 - \sqrt{3}\right)^{0} < 0$$
but
so the inequality has no solutions
$$3 \left(2 - \sqrt{3}\right)^{x} + \left(\sqrt{3} + 2\right)^{x} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$3 \left(2 - \sqrt{3}\right)^{x} + \left(\sqrt{3} + 2\right)^{x} = 0$$
Solve:
$$x_{1} = \frac{\log{\left(3 \right)} + i \pi}{\log{\left(4 \sqrt{3} + 7 \right)}}$$
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
$$\left(\sqrt{3} + 2\right)^{0} + 3 \left(2 - \sqrt{3}\right)^{0} < 0$$
4 < 0
but
4 > 0
so the inequality has no solutions
Solving inequality on a graph
Rapid solution
This inequality has no solutions