Mister Exam
x*(x-log(0.75)4)/log2(6x^2-11x+4)<=0 inequation
The solution
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x*(x - log(3/4)*4) ---------------------- <= 0 / / 2 \\ |log\6*x - 11*x + 4/| |--------------------| \ log(2) /
$$\frac{x \left(x - 4 \log{\left(\frac{3}{4} \right)}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(6 x^{2} - 11 x\right) + 4 \right)}} \leq 0$$
(x*(x - 4*log(3/4)))/((log(6*x^2 - 11*x + 4)/log(2))) <= 0
Detail solution
Given the inequality:
$$\frac{x \left(x - 4 \log{\left(\frac{3}{4} \right)}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(6 x^{2} - 11 x\right) + 4 \right)}} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x \left(x - 4 \log{\left(\frac{3}{4} \right)}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(6 x^{2} - 11 x\right) + 4 \right)}} = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = \log{\left(\frac{81}{256} \right)}$$
$$x_{1} = 0$$
$$x_{2} = \log{\left(\frac{81}{256} \right)}$$
This roots
$$x_{2} = \log{\left(\frac{81}{256} \right)}$$
$$x_{1} = 0$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\log{\left(\frac{81}{256} \right)} + - \frac{1}{10}$$
=
$$\log{\left(\frac{81}{256} \right)} - \frac{1}{10}$$
substitute to the expression
$$\frac{x \left(x - 4 \log{\left(\frac{3}{4} \right)}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(6 x^{2} - 11 x\right) + 4 \right)}} \leq 0$$
$$\frac{\left(\left(\log{\left(\frac{81}{256} \right)} - \frac{1}{10}\right) - 4 \log{\left(\frac{3}{4} \right)}\right) \left(\log{\left(\frac{81}{256} \right)} - \frac{1}{10}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 + \left(6 \left(\log{\left(\frac{81}{256} \right)} - \frac{1}{10}\right)^{2} - 11 \left(\log{\left(\frac{81}{256} \right)} - \frac{1}{10}\right)\right) \right)}} \leq 0$$
but
Then
$$x \leq \log{\left(\frac{81}{256} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq \log{\left(\frac{81}{256} \right)} \wedge x \leq 0$$
$$\frac{x \left(x - 4 \log{\left(\frac{3}{4} \right)}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(6 x^{2} - 11 x\right) + 4 \right)}} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{x \left(x - 4 \log{\left(\frac{3}{4} \right)}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(6 x^{2} - 11 x\right) + 4 \right)}} = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = \log{\left(\frac{81}{256} \right)}$$
$$x_{1} = 0$$
$$x_{2} = \log{\left(\frac{81}{256} \right)}$$
This roots
$$x_{2} = \log{\left(\frac{81}{256} \right)}$$
$$x_{1} = 0$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\log{\left(\frac{81}{256} \right)} + - \frac{1}{10}$$
=
$$\log{\left(\frac{81}{256} \right)} - \frac{1}{10}$$
substitute to the expression
$$\frac{x \left(x - 4 \log{\left(\frac{3}{4} \right)}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(\left(6 x^{2} - 11 x\right) + 4 \right)}} \leq 0$$
$$\frac{\left(\left(\log{\left(\frac{81}{256} \right)} - \frac{1}{10}\right) - 4 \log{\left(\frac{3}{4} \right)}\right) \left(\log{\left(\frac{81}{256} \right)} - \frac{1}{10}\right)}{\frac{1}{\log{\left(2 \right)}} \log{\left(4 + \left(6 \left(\log{\left(\frac{81}{256} \right)} - \frac{1}{10}\right)^{2} - 11 \left(\log{\left(\frac{81}{256} \right)} - \frac{1}{10}\right)\right) \right)}} \leq 0$$
/ 1 / 81\\ / 1 / 81\\
|- -- + log|---||*|- -- - 4*log(3/4) + log|---||*log(2)
\ 10 \256// \ 10 \256//
-------------------------------------------------------
/ 2\ <= 0
|51 / 81\ / 1 / 81\\ |
log|-- - 11*log|---| + 6*|- -- + log|---|| |
\10 \256/ \ 10 \256// /
but
/ 1 / 81\\ / 1 / 81\\
|- -- + log|---||*|- -- - 4*log(3/4) + log|---||*log(2)
\ 10 \256// \ 10 \256//
-------------------------------------------------------
/ 2\ >= 0
|51 / 81\ / 1 / 81\\ |
log|-- - 11*log|---| + 6*|- -- + log|---|| |
\10 \256/ \ 10 \256// /
Then
$$x \leq \log{\left(\frac{81}{256} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq \log{\left(\frac{81}{256} \right)} \wedge x \leq 0$$
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x2 x1
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