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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

$$x^{2} \log{\left(512 \right)} \geq \log{\left(2 \right)}$$

$$\left(-0.335702260395516\right)^{2} \log{\left(512 \right)} \geq \log{\left(2 \right)}$$

one of the solutions of our inequality is:

$$x \leq -0.235702260395516$$

Other solutions will get with the changeover to the next point

etc.

The answer:

$$x \leq -0.235702260395516$$

$$x \geq 0.235702260395516 \wedge x \leq 4$$

$$x^{2} \log{\left(512 \right)} \geq \log{\left(2 \right)}$$

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.

The answer:

$$x \leq -0.235702260395516$$

$$x \geq 0.235702260395516 \wedge x \leq 4$$

Solving inequality on a graph