Mister Exam
x-2\(5x-2)^2>0 inequation
The solution
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2
x - ---------- > 0
2
(5*x - 2)
$$x - \frac{2}{\left(5 x - 2\right)^{2}} > 0$$
x - 2/(5*x - 2)^2 > 0
Detail solution
Given the inequality:
$$x - \frac{2}{\left(5 x - 2\right)^{2}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x - \frac{2}{\left(5 x - 2\right)^{2}} = 0$$
Solve:
$$x_{1} = \frac{4}{15} + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}} + \frac{4}{225 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}}$$
$$x_{2} = \frac{4}{15} + \frac{4}{225 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
$$x_{3} = \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
Exclude the complex solutions:
$$x_{1} = \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
This roots
$$x_{1} = \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}\right)$$
=
$$\frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{1}{6} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
substitute to the expression
$$x - \frac{2}{\left(5 x - 2\right)^{2}} > 0$$
$$- \frac{2}{\left(-2 + 5 \left(\frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{1}{6} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}\right)\right)^{2}} + \left(\frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{1}{6} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}\right) > 0$$
Then
$$x < \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
no execute
the solution of our inequality is:
$$x > \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
$$x - \frac{2}{\left(5 x - 2\right)^{2}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x - \frac{2}{\left(5 x - 2\right)^{2}} = 0$$
Solve:
$$x_{1} = \frac{4}{15} + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}} + \frac{4}{225 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}}$$
$$x_{2} = \frac{4}{15} + \frac{4}{225 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
$$x_{3} = \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
Exclude the complex solutions:
$$x_{1} = \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
This roots
$$x_{1} = \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}\right)$$
=
$$\frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{1}{6} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
substitute to the expression
$$x - \frac{2}{\left(5 x - 2\right)^{2}} > 0$$
$$- \frac{2}{\left(-2 + 5 \left(\frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{1}{6} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}\right)\right)^{2}} + \left(\frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{1}{6} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}\right) > 0$$
_________________
/ ______
1 / 127 \/ 1785 2 4
- + 3 / ---- + -------- - ------------------------------------------------------------- + --------------------------
6 \/ 3375 1125 2 _________________
/ _________________ \ / ______
| / ______ | / 127 \/ 1785
| 7 / 127 \/ 1785 4 | 225*3 / ---- + -------- > 0
|- - + 5*3 / ---- + -------- + -------------------------| \/ 3375 1125
| 6 \/ 3375 1125 _________________|
| / ______ |
| / 127 \/ 1785 |
| 45*3 / ---- + -------- |
\ \/ 3375 1125 /
Then
$$x < \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
no execute
the solution of our inequality is:
$$x > \frac{4}{225 \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}} + \frac{4}{15} + \sqrt[3]{\frac{\sqrt{1785}}{1125} + \frac{127}{3375}}$$
_____
/
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x1
Solving inequality on a graph
Rapid solution
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/ 3 2 \ CRootOf\25*x - 20*x + 4*x - 2, 0/ < x
$$\operatorname{CRootOf} {\left(25 x^{3} - 20 x^{2} + 4 x - 2, 0\right)} < x$$
CRootOf(25*x^3 - 20*x^2 + 4*x - 2, 0) < x
Rapid solution 2
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/ 3 2 \ (CRootOf\25*x - 20*x + 4*x - 2, 0/, oo)
$$x\ in\ \left(\operatorname{CRootOf} {\left(25 x^{3} - 20 x^{2} + 4 x - 2, 0\right)}, \infty\right)$$
x in Interval.open(CRootOf(25*x^3 - 20*x^2 + 4*x - 2, 0), oo)