Mister Exam
x-5/(2x-1)^2>0 inequation
The solution
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5
x - ---------- > 0
2
(2*x - 1)
$$x - \frac{5}{\left(2 x - 1\right)^{2}} > 0$$
x - 5/(2*x - 1)^2 > 0
Detail solution
Given the inequality:
$$x - \frac{5}{\left(2 x - 1\right)^{2}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x - \frac{5}{\left(2 x - 1\right)^{2}} = 0$$
Solve:
$$x_{1} = \frac{1}{3} + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}} + \frac{1}{36 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}}$$
$$x_{2} = \frac{1}{3} + \frac{1}{36 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
$$x_{3} = \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
Exclude the complex solutions:
$$x_{1} = \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
This roots
$$x_{1} = \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}\right)$$
=
$$\frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{7}{30} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
substitute to the expression
$$x - \frac{5}{\left(2 x - 1\right)^{2}} > 0$$
$$- \frac{5}{\left(-1 + 2 \left(\frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{7}{30} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}\right)\right)^{2}} + \left(\frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{7}{30} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}\right) > 0$$
Then
$$x < \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
no execute
the solution of our inequality is:
$$x > \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
$$x - \frac{5}{\left(2 x - 1\right)^{2}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x - \frac{5}{\left(2 x - 1\right)^{2}} = 0$$
Solve:
$$x_{1} = \frac{1}{3} + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}} + \frac{1}{36 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}}$$
$$x_{2} = \frac{1}{3} + \frac{1}{36 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
$$x_{3} = \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
Exclude the complex solutions:
$$x_{1} = \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
This roots
$$x_{1} = \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}\right)$$
=
$$\frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{7}{30} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
substitute to the expression
$$x - \frac{5}{\left(2 x - 1\right)^{2}} > 0$$
$$- \frac{5}{\left(-1 + 2 \left(\frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{7}{30} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}\right)\right)^{2}} + \left(\frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{7}{30} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}\right) > 0$$
________________
/ ______
7 / 67 \/ 1995 5 1
-- + 3 / --- + -------- - ------------------------------------------------------------ + ------------------------
30 \/ 108 72 2 ________________
/ ________________ \ / ______
| / ______ | / 67 \/ 1995
| 8 / 67 \/ 1995 1 | 36*3 / --- + -------- > 0
|- -- + 2*3 / --- + -------- + ------------------------| \/ 108 72
| 15 \/ 108 72 ________________|
| / ______ |
| / 67 \/ 1995 |
| 18*3 / --- + -------- |
\ \/ 108 72 /
Then
$$x < \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
no execute
the solution of our inequality is:
$$x > \frac{1}{36 \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{1995}}{72} + \frac{67}{108}}$$
_____
/
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x1
Solving inequality on a graph
Rapid solution
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/ 3 2 \ CRootOf\4*x - 4*x + x - 5, 0/ < x
$$\operatorname{CRootOf} {\left(4 x^{3} - 4 x^{2} + x - 5, 0\right)} < x$$
CRootOf(4*x^3 - 4*x^2 + x - 5, 0) < x
Rapid solution 2
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/ 3 2 \ (CRootOf\4*x - 4*x + x - 5, 0/, oo)
$$x\ in\ \left(\operatorname{CRootOf} {\left(4 x^{3} - 4 x^{2} + x - 5, 0\right)}, \infty\right)$$
x in Interval.open(CRootOf(4*x^3 - 4*x^2 + x - 5, 0), oo)