Mister Exam
(x-4)x+5/x+2>0 inequation
The solution
You have entered
[src]
5
(x - 4)*x + - + 2 > 0
x
$$\left(x \left(x - 4\right) + \frac{5}{x}\right) + 2 > 0$$
x*(x - 4) + 5/x + 2 > 0
Detail solution
Given the inequality:
$$\left(x \left(x - 4\right) + \frac{5}{x}\right) + 2 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x \left(x - 4\right) + \frac{5}{x}\right) + 2 = 0$$
Solve:
$$x_{1} = \frac{4}{3} - \frac{10}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3}$$
$$x_{2} = \frac{4}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}$$
$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}$$
Exclude the complex solutions:
$$x_{1} = - \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}$$
This roots
$$x_{1} = - \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{37}{30}$$
substitute to the expression
$$\left(x \left(x - 4\right) + \frac{5}{x}\right) + 2 > 0$$
$$\left(\frac{5}{- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{37}{30}} + \left(-4 + \left(- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{37}{30}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{37}{30}\right)\right) + 2 > 0$$
the solution of our inequality is:
$$x < - \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}$$
$$\left(x \left(x - 4\right) + \frac{5}{x}\right) + 2 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x \left(x - 4\right) + \frac{5}{x}\right) + 2 = 0$$
Solve:
$$x_{1} = \frac{4}{3} - \frac{10}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3}$$
$$x_{2} = \frac{4}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}$$
$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}$$
Exclude the complex solutions:
$$x_{1} = - \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}$$
This roots
$$x_{1} = - \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{37}{30}$$
substitute to the expression
$$\left(x \left(x - 4\right) + \frac{5}{x}\right) + 2 > 0$$
$$\left(\frac{5}{- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{37}{30}} + \left(-4 + \left(- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{37}{30}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{37}{30}\right)\right) + 2 > 0$$
/ ________________\ / ________________\
| / _____ | | / _____ |
| / 79 3*\/ 249 | | / 79 3*\/ 249 |
| 3 / -- + --------- | | 3 / -- + --------- |
5 | 83 10 \/ 2 2 | |37 10 \/ 2 2 |
2 + ---------------------------------------------------- + |- -- - ----------------------- - ---------------------|*|-- - ----------------------- - ---------------------|
________________ | 30 ________________ 3 | |30 ________________ 3 |
/ _____ | / _____ | | / _____ |
/ 79 3*\/ 249 | / 79 3*\/ 249 | | / 79 3*\/ 249 | > 0
3 / -- + --------- | 3*3 / -- + --------- | | 3*3 / -- + --------- |
37 10 \/ 2 2 \ \/ 2 2 / \ \/ 2 2 /
-- - ----------------------- - ---------------------
30 ________________ 3
/ _____
/ 79 3*\/ 249
3*3 / -- + ---------
\/ 2 2 the solution of our inequality is:
$$x < - \frac{\sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}}{3} - \frac{10}{3 \sqrt[3]{\frac{3 \sqrt{249}}{2} + \frac{79}{2}}} + \frac{4}{3}$$
_____
\
-------ο-------
x1
Solving inequality on a graph
Rapid solution 2
[src]
/ 3 2 \ (-oo, CRootOf\x - 4*x + 2*x + 5, 0/) U (0, oo)
$$x\ in\ \left(-\infty, \operatorname{CRootOf} {\left(x^{3} - 4 x^{2} + 2 x + 5, 0\right)}\right) \cup \left(0, \infty\right)$$
x in Union(Interval.open(-oo, CRootOf(x^3 - 4*x^2 + 2*x + 5, 0)), Interval.open(0, oo))
Rapid solution
[src]
/ / 3 2 \\ Or\0 < x, x < CRootOf\x - 4*x + 2*x + 5, 0//
$$0 < x \vee x < \operatorname{CRootOf} {\left(x^{3} - 4 x^{2} + 2 x + 5, 0\right)}$$
(0 < x)∨(x < CRootOf(x^3 - 4*x^2 + 2*x + 5, 0))