Mister Exam
(log(x^2+9)/log(3))*log((4/5))*(3*x)/(x+2)-log((4/5))*(6-x)<=0 inequation
The solution
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/ 2 \
log\x + 9/
-----------*log(4/5)*3*x
log(3)
------------------------ - log(4/5)*(6 - x) <= 0
x + 2
$$\frac{3 x \frac{\log{\left(x^{2} + 9 \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{x + 2} - \left(6 - x\right) \log{\left(\frac{4}{5} \right)} \leq 0$$
((3*x)*((log(x^2 + 9)/log(3))*log(4/5)))/(x + 2) - (6 - x)*log(4/5) <= 0
Detail solution
Given the inequality:
$$\frac{3 x \frac{\log{\left(x^{2} + 9 \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{x + 2} - \left(6 - x\right) \log{\left(\frac{4}{5} \right)} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 x \frac{\log{\left(x^{2} + 9 \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{x + 2} - \left(6 - x\right) \log{\left(\frac{4}{5} \right)} = 0$$
Solve:
$$x_{1} = -10.0173582711649$$
$$x_{2} = 2.21834838848142$$
$$x_{1} = -10.0173582711649$$
$$x_{2} = 2.21834838848142$$
This roots
$$x_{1} = -10.0173582711649$$
$$x_{2} = 2.21834838848142$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-10.0173582711649 + - \frac{1}{10}$$
=
$$-10.1173582711649$$
substitute to the expression
$$\frac{3 x \frac{\log{\left(x^{2} + 9 \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{x + 2} - \left(6 - x\right) \log{\left(\frac{4}{5} \right)} \leq 0$$
$$\frac{\left(-10.1173582711649\right) 3 \frac{\log{\left(9 + \left(-10.1173582711649\right)^{2} \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{-10.1173582711649 + 2} - \left(6 - -10.1173582711649\right) \log{\left(\frac{4}{5} \right)} \leq 0$$
but
Then
$$x \leq -10.0173582711649$$
no execute
one of the solutions of our inequality is:
$$x \geq -10.0173582711649 \wedge x \leq 2.21834838848142$$
$$\frac{3 x \frac{\log{\left(x^{2} + 9 \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{x + 2} - \left(6 - x\right) \log{\left(\frac{4}{5} \right)} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3 x \frac{\log{\left(x^{2} + 9 \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{x + 2} - \left(6 - x\right) \log{\left(\frac{4}{5} \right)} = 0$$
Solve:
$$x_{1} = -10.0173582711649$$
$$x_{2} = 2.21834838848142$$
$$x_{1} = -10.0173582711649$$
$$x_{2} = 2.21834838848142$$
This roots
$$x_{1} = -10.0173582711649$$
$$x_{2} = 2.21834838848142$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-10.0173582711649 + - \frac{1}{10}$$
=
$$-10.1173582711649$$
substitute to the expression
$$\frac{3 x \frac{\log{\left(x^{2} + 9 \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{x + 2} - \left(6 - x\right) \log{\left(\frac{4}{5} \right)} \leq 0$$
$$\frac{\left(-10.1173582711649\right) 3 \frac{\log{\left(9 + \left(-10.1173582711649\right)^{2} \right)}}{\log{\left(3 \right)}} \log{\left(\frac{4}{5} \right)}}{-10.1173582711649 + 2} - \left(6 - -10.1173582711649\right) \log{\left(\frac{4}{5} \right)} \leq 0$$
17.6218104298214*log(4/5)
-16.1173582711649*log(4/5) + ------------------------- <= 0
log(3) but
17.6218104298214*log(4/5)
-16.1173582711649*log(4/5) + ------------------------- >= 0
log(3) Then
$$x \leq -10.0173582711649$$
no execute
one of the solutions of our inequality is:
$$x \geq -10.0173582711649 \wedge x \leq 2.21834838848142$$
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