Mister Exam
2log_2^2(x)-3log_2(x)+2<=0 inequation
The solution
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log(x) log(x) 2*------ - 3*------ + 2 <= 0 log(4) log(2)
$$\left(- 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right) + 2 \leq 0$$
-3*log(x)/log(2) + 2*(log(x)/log(4)) + 2 <= 0
Detail solution
Given the inequality:
$$\left(- 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right) + 2 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right) + 2 = 0$$
Solve:
$$x_{1} = 2$$
$$x_{1} = 2$$
This roots
$$x_{1} = 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$\left(- 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right) + 2 \leq 0$$
$$\left(- 3 \frac{\log{\left(\frac{19}{10} \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(\frac{19}{10} \right)}}{\log{\left(4 \right)}}\right) + 2 \leq 0$$
but
Then
$$x \leq 2$$
no execute
the solution of our inequality is:
$$x \geq 2$$
$$\left(- 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right) + 2 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right) + 2 = 0$$
Solve:
$$x_{1} = 2$$
$$x_{1} = 2$$
This roots
$$x_{1} = 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$\left(- 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(x \right)}}{\log{\left(4 \right)}}\right) + 2 \leq 0$$
$$\left(- 3 \frac{\log{\left(\frac{19}{10} \right)}}{\log{\left(2 \right)}} + 2 \frac{\log{\left(\frac{19}{10} \right)}}{\log{\left(4 \right)}}\right) + 2 \leq 0$$
/19\ /19\
3*log|--| 2*log|--|
\10/ \10/ <= 0
2 - --------- + ---------
log(2) log(4) but
/19\ /19\
3*log|--| 2*log|--|
\10/ \10/ >= 0
2 - --------- + ---------
log(2) log(4) Then
$$x \leq 2$$
no execute
the solution of our inequality is:
$$x \geq 2$$
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