Mister Exam
logx(x^2-14x+49)<0 inequation
The solution
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/ 2 \ log(x)*\x - 14*x + 49/ < 0
$$\left(\left(x^{2} - 14 x\right) + 49\right) \log{\left(x \right)} < 0$$
(x^2 - 14*x + 49)*log(x) < 0
Detail solution
Given the inequality:
$$\left(\left(x^{2} - 14 x\right) + 49\right) \log{\left(x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(x^{2} - 14 x\right) + 49\right) \log{\left(x \right)} = 0$$
Solve:
$$x_{1} = 1$$
$$x_{2} = 7$$
$$x_{1} = 1$$
$$x_{2} = 7$$
This roots
$$x_{1} = 1$$
$$x_{2} = 7$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1$$
=
$$\frac{9}{10}$$
substitute to the expression
$$\left(\left(x^{2} - 14 x\right) + 49\right) \log{\left(x \right)} < 0$$
$$\left(\left(- \frac{9 \cdot 14}{10} + \left(\frac{9}{10}\right)^{2}\right) + 49\right) \log{\left(\frac{9}{10} \right)} < 0$$
one of the solutions of our inequality is:
$$x < 1$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 1$$
$$x > 7$$
$$\left(\left(x^{2} - 14 x\right) + 49\right) \log{\left(x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(x^{2} - 14 x\right) + 49\right) \log{\left(x \right)} = 0$$
Solve:
$$x_{1} = 1$$
$$x_{2} = 7$$
$$x_{1} = 1$$
$$x_{2} = 7$$
This roots
$$x_{1} = 1$$
$$x_{2} = 7$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1$$
=
$$\frac{9}{10}$$
substitute to the expression
$$\left(\left(x^{2} - 14 x\right) + 49\right) \log{\left(x \right)} < 0$$
$$\left(\left(- \frac{9 \cdot 14}{10} + \left(\frac{9}{10}\right)^{2}\right) + 49\right) \log{\left(\frac{9}{10} \right)} < 0$$
3721*log(9/10)
-------------- < 0
100 one of the solutions of our inequality is:
$$x < 1$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 1$$
$$x > 7$$
Solving inequality on a graph