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log(x)*(4*x - 3) >= 2

$$\left(4 x - 3\right) \log{\left(x \right)} \geq 2$$

(4*x - 1*3)*log(x) >= 2

Detail solution

Given the inequality:

$$\left(4 x - 3\right) \log{\left(x \right)} \geq 2$$

To solve this inequality, we must first solve the corresponding equation:

$$\left(4 x - 3\right) \log{\left(x \right)} = 2$$

Solve:

$$x_{1} = 1.69623401783029$$

$$x_{1} = 1.69623401783029$$

This roots

$$x_{1} = 1.69623401783029$$

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

$$x_{0} = x_{1} - \frac{1}{10}$$

=

$$- \frac{1}{10} + 1.69623401783029$$

=

$$1.59623401783029$$

substitute to the expression

$$\left(4 x - 3\right) \log{\left(x \right)} \geq 2$$

$$\left(\left(-1\right) 3 + 4 \cdot 1.59623401783029\right) \log{\left(1.59623401783029 \right)} \geq 2$$

but

Then

$$x \leq 1.69623401783029$$

no execute

the solution of our inequality is:

$$x \geq 1.69623401783029$$

$$\left(4 x - 3\right) \log{\left(x \right)} \geq 2$$

To solve this inequality, we must first solve the corresponding equation:

$$\left(4 x - 3\right) \log{\left(x \right)} = 2$$

Solve:

$$x_{1} = 1.69623401783029$$

$$x_{1} = 1.69623401783029$$

This roots

$$x_{1} = 1.69623401783029$$

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

$$x_{0} = x_{1} - \frac{1}{10}$$

=

$$- \frac{1}{10} + 1.69623401783029$$

=

$$1.59623401783029$$

substitute to the expression

$$\left(4 x - 3\right) \log{\left(x \right)} \geq 2$$

$$\left(\left(-1\right) 3 + 4 \cdot 1.59623401783029\right) \log{\left(1.59623401783029 \right)} \geq 2$$

1.58295559157076 >= 2

but

1.58295559157076 < 2

Then

$$x \leq 1.69623401783029$$

no execute

the solution of our inequality is:

$$x \geq 1.69623401783029$$

_____ / -------•------- x_1

Solving inequality on a graph