Mister Exam
2/log2(2x-2)+3/log2(4x-4)<=8/log3(27)+log2(x-1) inequation
The solution
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2 3 8 log(x - 1) -------------- + -------------- <= --------- + ---------- /log(2*x - 2)\ /log(4*x - 4)\ /log(27)\ log(2) |------------| |------------| |-------| \ log(2) / \ log(2) / \ log(3)/
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 x - 4 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(2 x - 2 \right)}} \leq \frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
3/((log(4*x - 1*4)/log(2))) + 2/((log(2*x - 1*2)/log(2))) <= log(x - 1*1)/log(2) + 8/((log(27)/log(3)))
Detail solution
Given the inequality:
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 x - 4 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(2 x - 2 \right)}} \leq \frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 x - 4 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(2 x - 2 \right)}} = \frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
Solve:
$$x_{1} = 2.19401177221178$$
$$x_{1} = 2.19401177221178$$
This roots
$$x_{1} = 2.19401177221178$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2.19401177221178$$
=
$$2.09401177221178$$
substitute to the expression
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 x - 4 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(2 x - 2 \right)}} \leq \frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(-1\right) 4 + 4 \cdot 2.09401177221178 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(-1\right) 2 + 2 \cdot 2.09401177221178 \right)}} \leq \frac{\log{\left(\left(-1\right) 1 + 2.09401177221178 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
but
Then
$$x \leq 2.19401177221178$$
no execute
the solution of our inequality is:
$$x \geq 2.19401177221178$$
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 x - 4 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(2 x - 2 \right)}} \leq \frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 x - 4 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(2 x - 2 \right)}} = \frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
Solve:
$$x_{1} = 2.19401177221178$$
$$x_{1} = 2.19401177221178$$
This roots
$$x_{1} = 2.19401177221178$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2.19401177221178$$
=
$$2.09401177221178$$
substitute to the expression
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 x - 4 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(2 x - 2 \right)}} \leq \frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
$$\frac{3}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(-1\right) 4 + 4 \cdot 2.09401177221178 \right)}} + \frac{2}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(-1\right) 2 + 2 \cdot 2.09401177221178 \right)}} \leq \frac{\log{\left(\left(-1\right) 1 + 2.09401177221178 \right)}}{\log{\left(2 \right)}} + \frac{8}{\frac{1}{\log{\left(3 \right)}} \log{\left(27 \right)}}$$
0.0898514646473563 8*log(3)
4.58660237212683*log(2) <= ------------------ + --------
log(2) log(27) but
0.0898514646473563 8*log(3)
4.58660237212683*log(2) >= ------------------ + --------
log(2) log(27) Then
$$x \leq 2.19401177221178$$
no execute
the solution of our inequality is:
$$x \geq 2.19401177221178$$
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x_1
Solving inequality on a graph