Mister Exam

# x^2*log(512,(4-x))>=log(2,(x^2-8*x+16)) inequation

A inequation with variable

### The solution

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 2                       /    2           \
x *log(512, 4 - x) >= log\2, x  - 8*x + 16/
$$x^{2} \log{\left(512 \right)} \geq \log{\left(2 \right)}$$
x^2*log(512, 4 - x) >= log(2, x^2 - 8*x + 16)
Detail solution
Given the inequality:
$$x^{2} \log{\left(512 \right)} \geq \log{\left(2 \right)}$$
To solve this inequality, we must first solve the corresponding equation:
$$x^{2} \log{\left(512 \right)} = \log{\left(2 \right)}$$
Solve:
$$x_{1} = -0.235702260395516$$
$$x_{2} = 0.235702260395516$$
$$x_{3} = \mathtt{\text{(0.23570226039551564-2.4864463269855524e-16j)}}$$
$$x_{4} = \mathtt{\text{(-0.23570226039551584+5.5791818189189e-18j)}}$$
$$x_{5} = \mathtt{\text{(0.23570226039551584-5.447826629452265e-18j)}}$$
$$x_{6} = \mathtt{\text{(0.2357022603955008-2.316336302886255e-14j)}}$$
$$x_{7} = 4$$
Exclude the complex solutions:
$$x_{1} = -0.235702260395516$$
$$x_{2} = 0.235702260395516$$
$$x_{3} = 4$$
This roots
$$x_{1} = -0.235702260395516$$
$$x_{2} = 0.235702260395516$$
$$x_{3} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-0.235702260395516 - \frac{1}{10}$$
=
$$-0.335702260395516$$
substitute to the expression
$$x^{2} \log{\left(512 \right)} \geq \log{\left(2 \right)}$$
$$\left(-0.335702260395516\right)^{2} \log{\left(512 \right)} \geq \log{\left(2 \right)}$$
0.0768268239052697*log(512) >= 0.340858675998218*log(2)

one of the solutions of our inequality is:
$$x \leq -0.235702260395516$$
 _____           _____
\         /     \
-------•-------•-------•-------
x_1      x_2      x_3

Other solutions will get with the changeover to the next point
etc.
$$x \leq -0.235702260395516$$
$$x \geq 0.235702260395516 \wedge x \leq 4$$