Mister Exam
lg(3x+2*sqrt(x)-1)/lg((5x+3*sqrt(x)-2)^5)>=(log(11)/log(32))/(log(11)/log(2)) inequation
The solution
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/log(11)\
/ ___ \ |-------|
log\3*x + 2*\/ x - 1/ \log(32)/
------------------------- >= ---------
/ 5\ /log(11)\
|/ ___ \ | |-------|
log\\5*x + 3*\/ x - 2/ / \ log(2)/
$$\frac{\log{\left(\left(2 \sqrt{x} + 3 x\right) - 1 \right)}}{\log{\left(\left(\left(3 \sqrt{x} + 5 x\right) - 2\right)^{5} \right)}} \geq \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
log(2*sqrt(x) + 3*x - 1)/log((3*sqrt(x) + 5*x - 2)^5) >= (log(11)/log(32))/((log(11)/log(2)))
Detail solution
Given the inequality:
$$\frac{\log{\left(\left(2 \sqrt{x} + 3 x\right) - 1 \right)}}{\log{\left(\left(\left(3 \sqrt{x} + 5 x\right) - 2\right)^{5} \right)}} \geq \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(\left(2 \sqrt{x} + 3 x\right) - 1 \right)}}{\log{\left(\left(\left(3 \sqrt{x} + 5 x\right) - 2\right)^{5} \right)}} = \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
Solve:
$$x_{1} = 0.25$$
$$x_{1} = 0.25$$
This roots
$$x_{1} = 0.25$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.25$$
=
$$0.15$$
substitute to the expression
$$\frac{\log{\left(\left(2 \sqrt{x} + 3 x\right) - 1 \right)}}{\log{\left(\left(\left(3 \sqrt{x} + 5 x\right) - 2\right)^{5} \right)}} \geq \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
$$\frac{\log{\left(-1 + \left(0.15 \cdot 3 + 2 \sqrt{0.15}\right) \right)}}{\log{\left(\left(-2 + \left(0.15 \cdot 5 + 3 \sqrt{0.15}\right)\right)^{5} \right)}} \geq \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
Then
$$x \leq 0.25$$
no execute
the solution of our inequality is:
$$x \geq 0.25$$
$$\frac{\log{\left(\left(2 \sqrt{x} + 3 x\right) - 1 \right)}}{\log{\left(\left(\left(3 \sqrt{x} + 5 x\right) - 2\right)^{5} \right)}} \geq \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(\left(2 \sqrt{x} + 3 x\right) - 1 \right)}}{\log{\left(\left(\left(3 \sqrt{x} + 5 x\right) - 2\right)^{5} \right)}} = \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
Solve:
$$x_{1} = 0.25$$
$$x_{1} = 0.25$$
This roots
$$x_{1} = 0.25$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.25$$
=
$$0.15$$
substitute to the expression
$$\frac{\log{\left(\left(2 \sqrt{x} + 3 x\right) - 1 \right)}}{\log{\left(\left(\left(3 \sqrt{x} + 5 x\right) - 2\right)^{5} \right)}} \geq \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
$$\frac{\log{\left(-1 + \left(0.15 \cdot 3 + 2 \sqrt{0.15}\right) \right)}}{\log{\left(\left(-2 + \left(0.15 \cdot 5 + 3 \sqrt{0.15}\right)\right)^{5} \right)}} \geq \frac{\log{\left(11 \right)} \frac{1}{\log{\left(32 \right)}}}{\frac{1}{\log{\left(2 \right)}} \log{\left(11 \right)}}$$
-1.49344906652291 log(2) ------------------------ >= ------- -12.1461301889903 + pi*I log(32)
Then
$$x \leq 0.25$$
no execute
the solution of our inequality is:
$$x \geq 0.25$$
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Solving inequality on a graph