Mister Exam
log((x-8)^8/(x-1))<=8 inequation
The solution
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/ 8\ |(x - 8) | log|--------| <= 8 \ x - 1 /
$$\log{\left(\frac{\left(x - 8\right)^{8}}{x - 1} \right)} \leq 8$$
log((x - 8)^8/(x - 1)) <= 8
Detail solution
Given the inequality:
$$\log{\left(\frac{\left(x - 8\right)^{8}}{x - 1} \right)} \leq 8$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{\left(x - 8\right)^{8}}{x - 1} \right)} = 8$$
Solve:
$$x_{1} = 4.78920221490978$$
$$x_{2} = 11.6536925692719$$
$$x_{1} = 4.78920221490978$$
$$x_{2} = 11.6536925692719$$
This roots
$$x_{1} = 4.78920221490978$$
$$x_{2} = 11.6536925692719$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 4.78920221490978$$
=
$$4.68920221490978$$
substitute to the expression
$$\log{\left(\frac{\left(x - 8\right)^{8}}{x - 1} \right)} \leq 8$$
$$\log{\left(\frac{\left(-8 + 4.68920221490978\right)^{8}}{-1 + 4.68920221490978} \right)} \leq 8$$
but
Then
$$x \leq 4.78920221490978$$
no execute
one of the solutions of our inequality is:
$$x \geq 4.78920221490978 \wedge x \leq 11.6536925692719$$
$$\log{\left(\frac{\left(x - 8\right)^{8}}{x - 1} \right)} \leq 8$$
To solve this inequality, we must first solve the corresponding equation:
$$\log{\left(\frac{\left(x - 8\right)^{8}}{x - 1} \right)} = 8$$
Solve:
$$x_{1} = 4.78920221490978$$
$$x_{2} = 11.6536925692719$$
$$x_{1} = 4.78920221490978$$
$$x_{2} = 11.6536925692719$$
This roots
$$x_{1} = 4.78920221490978$$
$$x_{2} = 11.6536925692719$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 4.78920221490978$$
=
$$4.68920221490978$$
substitute to the expression
$$\log{\left(\frac{\left(x - 8\right)^{8}}{x - 1} \right)} \leq 8$$
$$\log{\left(\frac{\left(-8 + 4.68920221490978\right)^{8}}{-1 + 4.68920221490978} \right)} \leq 8$$
8.27210323151234 <= 8
but
8.27210323151234 >= 8
Then
$$x \leq 4.78920221490978$$
no execute
one of the solutions of our inequality is:
$$x \geq 4.78920221490978 \wedge x \leq 11.6536925692719$$
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