Mister Exam
lg(1-4/(x+4))+(lg(2/(x-1)-1)^2)/lg(1+(4/x))<0 inequation
The solution
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$$\frac{\log{\left(-1 + \frac{2}{x - 1} \right)}^{2}}{\log{\left(1 + \frac{4}{x} \right)}} + \log{\left(1 - \frac{4}{x + 4} \right)} < 0$$
log(-1 + 2/(x - 1))^2/log(1 + 4/x) + log(1 - 4/(x + 4)) < 0
Detail solution
Given the inequality:
$$\frac{\log{\left(-1 + \frac{2}{x - 1} \right)}^{2}}{\log{\left(1 + \frac{4}{x} \right)}} + \log{\left(1 - \frac{4}{x + 4} \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(-1 + \frac{2}{x - 1} \right)}^{2}}{\log{\left(1 + \frac{4}{x} \right)}} + \log{\left(1 - \frac{4}{x + 4} \right)} = 0$$
Solve:
$$x_{1} = -1.4142135623731$$
$$x_{2} = 2.44948974278318$$
$$x_{1} = -1.4142135623731$$
$$x_{2} = 2.44948974278318$$
This roots
$$x_{1} = -1.4142135623731$$
$$x_{2} = 2.44948974278318$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1.4142135623731 + - \frac{1}{10}$$
=
$$-1.5142135623731$$
substitute to the expression
$$\frac{\log{\left(-1 + \frac{2}{x - 1} \right)}^{2}}{\log{\left(1 + \frac{4}{x} \right)}} + \log{\left(1 - \frac{4}{x + 4} \right)} < 0$$
$$\frac{\log{\left(-1 + \frac{2}{-1.5142135623731 - 1} \right)}^{2}}{\log{\left(\frac{4}{-1.5142135623731} + 1 \right)}} + \log{\left(1 - \frac{4}{-1.5142135623731 + 4} \right)} < 0$$
Then
$$x < -1.4142135623731$$
no execute
one of the solutions of our inequality is:
$$x > -1.4142135623731 \wedge x < 2.44948974278318$$
$$\frac{\log{\left(-1 + \frac{2}{x - 1} \right)}^{2}}{\log{\left(1 + \frac{4}{x} \right)}} + \log{\left(1 - \frac{4}{x + 4} \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(-1 + \frac{2}{x - 1} \right)}^{2}}{\log{\left(1 + \frac{4}{x} \right)}} + \log{\left(1 - \frac{4}{x + 4} \right)} = 0$$
Solve:
$$x_{1} = -1.4142135623731$$
$$x_{2} = 2.44948974278318$$
$$x_{1} = -1.4142135623731$$
$$x_{2} = 2.44948974278318$$
This roots
$$x_{1} = -1.4142135623731$$
$$x_{2} = 2.44948974278318$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1.4142135623731 + - \frac{1}{10}$$
=
$$-1.5142135623731$$
substitute to the expression
$$\frac{\log{\left(-1 + \frac{2}{x - 1} \right)}^{2}}{\log{\left(1 + \frac{4}{x} \right)}} + \log{\left(1 - \frac{4}{x + 4} \right)} < 0$$
$$\frac{\log{\left(-1 + \frac{2}{-1.5142135623731 - 1} \right)}^{2}}{\log{\left(\frac{4}{-1.5142135623731} + 1 \right)}} + \log{\left(1 - \frac{4}{-1.5142135623731 + 4} \right)} < 0$$
2
(0.585270932597601 + pi*I)
-0.495692879937354 + pi*I + --------------------------- < 0
0.495692879937354 + pi*I
Then
$$x < -1.4142135623731$$
no execute
one of the solutions of our inequality is:
$$x > -1.4142135623731 \wedge x < 2.44948974278318$$
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