Mister Exam
x^3-8*x-9>=0 inequation
The solution
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3 x - 8*x - 9 >= 0
$$\left(x^{3} - 8 x\right) - 9 \geq 0$$
x^3 - 8*x - 9 >= 0
Detail solution
Given the inequality:
$$\left(x^{3} - 8 x\right) - 9 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{3} - 8 x\right) - 9 = 0$$
Solve:
$$x_{1} = \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}} + \frac{8}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}}$$
$$x_{2} = \frac{8}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
$$x_{3} = \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
Exclude the complex solutions:
$$x_{1} = \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
This roots
$$x_{1} = \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}\right)$$
=
$$- \frac{1}{10} + \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
substitute to the expression
$$\left(x^{3} - 8 x\right) - 9 \geq 0$$
$$-9 + \left(- 8 \left(- \frac{1}{10} + \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}\right) + \left(- \frac{1}{10} + \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}\right)^{3}\right) \geq 0$$
but
Then
$$x \leq \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
no execute
the solution of our inequality is:
$$x \geq \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
$$\left(x^{3} - 8 x\right) - 9 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{3} - 8 x\right) - 9 = 0$$
Solve:
$$x_{1} = \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}} + \frac{8}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}}$$
$$x_{2} = \frac{8}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
$$x_{3} = \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
Exclude the complex solutions:
$$x_{1} = \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
This roots
$$x_{1} = \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}\right)$$
=
$$- \frac{1}{10} + \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
substitute to the expression
$$\left(x^{3} - 8 x\right) - 9 \geq 0$$
$$-9 + \left(- 8 \left(- \frac{1}{10} + \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}\right) + \left(- \frac{1}{10} + \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}\right)^{3}\right) \geq 0$$
3
/ _____________ \ _____________
| / _____ | / _____
41 | 1 / 9 \/ 417 8 | / 9 \/ 417 64
- -- + |- -- + 3 / - + ------- + --------------------| - 8*3 / - + ------- - --------------------
5 | 10 \/ 2 18 _____________| \/ 2 18 _____________ >= 0
| / _____ | / _____
| / 9 \/ 417 | / 9 \/ 417
| 3*3 / - + ------- | 3*3 / - + -------
\ \/ 2 18 / \/ 2 18
but
3
/ _____________ \ _____________
| / _____ | / _____
41 | 1 / 9 \/ 417 8 | / 9 \/ 417 64
- -- + |- -- + 3 / - + ------- + --------------------| - 8*3 / - + ------- - --------------------
5 | 10 \/ 2 18 _____________| \/ 2 18 _____________ < 0
| / _____ | / _____
| / 9 \/ 417 | / 9 \/ 417
| 3*3 / - + ------- | 3*3 / - + -------
\ \/ 2 18 / \/ 2 18
Then
$$x \leq \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
no execute
the solution of our inequality is:
$$x \geq \frac{8}{3 \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{417}}{18} + \frac{9}{2}}$$
_____
/
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x1
Solving inequality on a graph
Rapid solution 2
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/ 3 \ [CRootOf\x - 8*x - 9, 0/, oo)
$$x\ in\ \left[\operatorname{CRootOf} {\left(x^{3} - 8 x - 9, 0\right)}, \infty\right)$$
x in Interval(CRootOf(x^3 - 8*x - 9, 0), oo)
Rapid solution
[src]
/ / 3 \ \ And\CRootOf\x - 8*x - 9, 0/ <= x, x < oo/
$$\operatorname{CRootOf} {\left(x^{3} - 8 x - 9, 0\right)} \leq x \wedge x < \infty$$
(x < oo)∧(CRootOf(x^3 - 8*x - 9, 0) <= x)