Mister Exam
3*lg^2(x-8)/(lg^2(x-4))>=2 inequation
The solution
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2
3*log (x - 8)
------------- >= 2
2
log (x - 4)
$$\frac{3 \log{\left(x - 8 \right)}^{2}}{\log{\left(x - 4 \right)}^{2}} \geq 2$$
3*log(x - 1*8)^2/(log(x - 1*4)^2) >= 2
Detail solution
Given the inequality:
$$\frac{3 \log{\left(x - 8 \right)}^{2}}{\log{\left(x - 4 \right)}^{2}} \geq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 \log{\left(x - 8 \right)}^{2}}{\log{\left(x - 4 \right)}^{2}} = 2$$
Solve:
$$x_{1} = 15.1773834231572$$
$$x_{1} = 15.1773834231572$$
This roots
$$x_{1} = 15.1773834231572$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 15.1773834231572$$
=
$$15.0773834231572$$
substitute to the expression
$$\frac{3 \log{\left(x - 8 \right)}^{2}}{\log{\left(x - 4 \right)}^{2}} \geq 2$$
$$\frac{3 \log{\left(\left(-1\right) 8 + 15.0773834231572 \right)}^{2}}{\log{\left(\left(-1\right) 4 + 15.0773834231572 \right)}^{2}} \geq 2$$
but
Then
$$x \leq 15.1773834231572$$
no execute
the solution of our inequality is:
$$x \geq 15.1773834231572$$
$$\frac{3 \log{\left(x - 8 \right)}^{2}}{\log{\left(x - 4 \right)}^{2}} \geq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 \log{\left(x - 8 \right)}^{2}}{\log{\left(x - 4 \right)}^{2}} = 2$$
Solve:
$$x_{1} = 15.1773834231572$$
$$x_{1} = 15.1773834231572$$
This roots
$$x_{1} = 15.1773834231572$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 15.1773834231572$$
=
$$15.0773834231572$$
substitute to the expression
$$\frac{3 \log{\left(x - 8 \right)}^{2}}{\log{\left(x - 4 \right)}^{2}} \geq 2$$
$$\frac{3 \log{\left(\left(-1\right) 8 + 15.0773834231572 \right)}^{2}}{\log{\left(\left(-1\right) 4 + 15.0773834231572 \right)}^{2}} \geq 2$$
1.98638937528934 >= 2
but
1.98638937528934 < 2
Then
$$x \leq 15.1773834231572$$
no execute
the solution of our inequality is:
$$x \geq 15.1773834231572$$
_____
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