Mister Exam
(3*x+7)*log2x+5(x^2+4*x+5)>=0 inequation
The solution
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/ 2 \ (3*x + 7)*log(2*x) + 5*\x + 4*x + 5/ >= 0
$$\left(3 x + 7\right) \log{\left(2 x \right)} + 5 \left(\left(x^{2} + 4 x\right) + 5\right) \geq 0$$
(3*x + 7)*log(2*x) + 5*(x^2 + 4*x + 5) >= 0
Detail solution
Given the inequality:
$$\left(3 x + 7\right) \log{\left(2 x \right)} + 5 \left(\left(x^{2} + 4 x\right) + 5\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(3 x + 7\right) \log{\left(2 x \right)} + 5 \left(\left(x^{2} + 4 x\right) + 5\right) = 0$$
Solve:
$$x_{1} = 0.0138024016738796$$
$$x_{2} = -2.38370857245839 + 0.488201789181448 i$$
$$x_{3} = -2.38370857245839 - 0.488201789181448 i$$
Exclude the complex solutions:
$$x_{1} = 0.0138024016738796$$
This roots
$$x_{1} = 0.0138024016738796$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.0138024016738796$$
=
$$-0.0861975983261204$$
substitute to the expression
$$\left(3 x + 7\right) \log{\left(2 x \right)} + 5 \left(\left(x^{2} + 4 x\right) + 5\right) \geq 0$$
$$5 \left(\left(\left(-0.0861975983261204\right) 4 + \left(-0.0861975983261204\right)^{2}\right) + 5\right) + \left(\left(-0.0861975983261204\right) 3 + 7\right) \log{\left(\left(-0.0861975983261204\right) 2 \right)} \geq 0$$
Then
$$x \leq 0.0138024016738796$$
no execute
the solution of our inequality is:
$$x \geq 0.0138024016738796$$
$$\left(3 x + 7\right) \log{\left(2 x \right)} + 5 \left(\left(x^{2} + 4 x\right) + 5\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(3 x + 7\right) \log{\left(2 x \right)} + 5 \left(\left(x^{2} + 4 x\right) + 5\right) = 0$$
Solve:
$$x_{1} = 0.0138024016738796$$
$$x_{2} = -2.38370857245839 + 0.488201789181448 i$$
$$x_{3} = -2.38370857245839 - 0.488201789181448 i$$
Exclude the complex solutions:
$$x_{1} = 0.0138024016738796$$
This roots
$$x_{1} = 0.0138024016738796$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.0138024016738796$$
=
$$-0.0861975983261204$$
substitute to the expression
$$\left(3 x + 7\right) \log{\left(2 x \right)} + 5 \left(\left(x^{2} + 4 x\right) + 5\right) \geq 0$$
$$5 \left(\left(\left(-0.0861975983261204\right) 4 + \left(-0.0861975983261204\right)^{2}\right) + 5\right) + \left(\left(-0.0861975983261204\right) 3 + 7\right) \log{\left(\left(-0.0861975983261204\right) 2 \right)} \geq 0$$
11.4620349690043 + 6.74140720502164*pi*I >= 0
Then
$$x \leq 0.0138024016738796$$
no execute
the solution of our inequality is:
$$x \geq 0.0138024016738796$$
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