Mister Exam
thex<√3/3 inequation
The solution
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/ x\ \/ 3
tanh\E / < -----
3
$$\tanh{\left(e^{x} \right)} < \frac{\sqrt{3}}{3}$$
tanh(E^x) < sqrt(3)/3
Detail solution
Given the inequality:
$$\tanh{\left(e^{x} \right)} < \frac{\sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tanh{\left(e^{x} \right)} = \frac{\sqrt{3}}{3}$$
Solve:
$$x_{1} = \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} + i \pi \right)}$$
$$x_{2} = \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)}$$
Exclude the complex solutions:
$$x_{1} = \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)}$$
This roots
$$x_{1} = \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)} + - \frac{1}{10}$$
=
$$\log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)} - \frac{1}{10}$$
substitute to the expression
$$\tanh{\left(e^{x} \right)} < \frac{\sqrt{3}}{3}$$
$$\tanh{\left(e^{\log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)} - \frac{1}{10}} \right)} < \frac{\sqrt{3}}{3}$$
the solution of our inequality is:
$$x < \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)}$$
$$\tanh{\left(e^{x} \right)} < \frac{\sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tanh{\left(e^{x} \right)} = \frac{\sqrt{3}}{3}$$
Solve:
$$x_{1} = \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} + i \pi \right)}$$
$$x_{2} = \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)}$$
Exclude the complex solutions:
$$x_{1} = \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)}$$
This roots
$$x_{1} = \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)} + - \frac{1}{10}$$
=
$$\log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)} - \frac{1}{10}$$
substitute to the expression
$$\tanh{\left(e^{x} \right)} < \frac{\sqrt{3}}{3}$$
$$\tanh{\left(e^{\log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)} - \frac{1}{10}} \right)} < \frac{\sqrt{3}}{3}$$
/ / ____________ ___________\\ ___
| -1/10 | / -1 / ___ || \/ 3
tanh|e *log| / ---------- *\/ 3 + \/ 3 || < -----
| | / ___ || 3
\ \\/ -3 + \/ 3 // the solution of our inequality is:
$$x < \log{\left(\log{\left(\sqrt{- \frac{1}{-3 + \sqrt{3}}} \sqrt{\sqrt{3} + 3} \right)} \right)}$$
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x1
Solving inequality on a graph