Mister Exam
logx(x^2+3x-3)>1 inequation
The solution
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/ 2 \ log(x)*\x + 3*x - 3/ > 1
$$\left(x^{2} + 3 x - 3\right) \log{\left(x \right)} > 1$$
(x^2 + 3*x - 1*3)*log(x) > 1
Detail solution
Given the inequality:
$$\left(x^{2} + 3 x - 3\right) \log{\left(x \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{2} + 3 x - 3\right) \log{\left(x \right)} = 1$$
Solve:
$$x_{1} = 1.38476818988409$$
$$x_{2} = \mathtt{\text{(-3.8171534623893013+0.05831405799227356j)}}$$
Exclude the complex solutions:
$$x_{1} = 1.38476818988409$$
This roots
$$x_{1} = 1.38476818988409$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.38476818988409$$
=
$$1.28476818988409$$
substitute to the expression
$$\left(x^{2} + 3 x - 3\right) \log{\left(x \right)} > 1$$
$$\left(\left(-1\right) 3 + 1.28476818988409^{2} + 3 \cdot 1.28476818988409\right) \log{\left(1.28476818988409 \right)} > 1$$
Then
$$x < 1.38476818988409$$
no execute
the solution of our inequality is:
$$x > 1.38476818988409$$
$$\left(x^{2} + 3 x - 3\right) \log{\left(x \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{2} + 3 x - 3\right) \log{\left(x \right)} = 1$$
Solve:
$$x_{1} = 1.38476818988409$$
$$x_{2} = \mathtt{\text{(-3.8171534623893013+0.05831405799227356j)}}$$
Exclude the complex solutions:
$$x_{1} = 1.38476818988409$$
This roots
$$x_{1} = 1.38476818988409$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1.38476818988409$$
=
$$1.28476818988409$$
substitute to the expression
$$\left(x^{2} + 3 x - 3\right) \log{\left(x \right)} > 1$$
$$\left(\left(-1\right) 3 + 1.28476818988409^{2} + 3 \cdot 1.28476818988409\right) \log{\left(1.28476818988409 \right)} > 1$$
0.627682083871368 > 1
Then
$$x < 1.38476818988409$$
no execute
the solution of our inequality is:
$$x > 1.38476818988409$$
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