Mister Exam
log(x)*(3*x-1)/(x*x+1)>0 inequation
The solution
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[src]
log(x)*(3*x - 1)
---------------- > 0
x*x + 1
$$\frac{\left(3 x - 1\right) \log{\left(x \right)}}{x x + 1} > 0$$
((3*x - 1)*log(x))/(x*x + 1) > 0
Detail solution
Given the inequality:
$$\frac{\left(3 x - 1\right) \log{\left(x \right)}}{x x + 1} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(3 x - 1\right) \log{\left(x \right)}}{x x + 1} = 0$$
Solve:
$$x_{1} = \frac{1}{3}$$
$$x_{2} = 1$$
$$x_{1} = \frac{1}{3}$$
$$x_{2} = 1$$
This roots
$$x_{1} = \frac{1}{3}$$
$$x_{2} = 1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{1}{3}$$
=
$$\frac{7}{30}$$
substitute to the expression
$$\frac{\left(3 x - 1\right) \log{\left(x \right)}}{x x + 1} > 0$$
$$\frac{\left(-1 + \frac{3 \cdot 7}{30}\right) \log{\left(\frac{7}{30} \right)}}{\frac{7 \cdot 7}{30 \cdot 30} + 1} > 0$$
one of the solutions of our inequality is:
$$x < \frac{1}{3}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{1}{3}$$
$$x > 1$$
$$\frac{\left(3 x - 1\right) \log{\left(x \right)}}{x x + 1} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(3 x - 1\right) \log{\left(x \right)}}{x x + 1} = 0$$
Solve:
$$x_{1} = \frac{1}{3}$$
$$x_{2} = 1$$
$$x_{1} = \frac{1}{3}$$
$$x_{2} = 1$$
This roots
$$x_{1} = \frac{1}{3}$$
$$x_{2} = 1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{1}{3}$$
=
$$\frac{7}{30}$$
substitute to the expression
$$\frac{\left(3 x - 1\right) \log{\left(x \right)}}{x x + 1} > 0$$
$$\frac{\left(-1 + \frac{3 \cdot 7}{30}\right) \log{\left(\frac{7}{30} \right)}}{\frac{7 \cdot 7}{30 \cdot 30} + 1} > 0$$
-270*log(7/30)
-------------- > 0
949 one of the solutions of our inequality is:
$$x < \frac{1}{3}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{1}{3}$$
$$x > 1$$
Solving inequality on a graph
Rapid solution
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Or(And(0 < x, x < 1/3), And(1 < x, x < oo))
$$\left(0 < x \wedge x < \frac{1}{3}\right) \vee \left(1 < x \wedge x < \infty\right)$$
((0 < x)∧(x < 1/3))∨((1 < x)∧(x < oo))
Rapid solution 2
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(0, 1/3) U (1, oo)
$$x\ in\ \left(0, \frac{1}{3}\right) \cup \left(1, \infty\right)$$
x in Union(Interval.open(0, 1/3), Interval.open(1, oo))