Mister Exam
(((x^2)+5*x+7)^2)-(x+2)*(x-3)<1 inequation
The solution
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2 / 2 \ \x + 5*x + 7/ - (x + 2)*(x - 3) < 1
$$- \left(x - 3\right) \left(x + 2\right) + \left(\left(x^{2} + 5 x\right) + 7\right)^{2} < 1$$
-(x - 3)*(x + 2) + (x^2 + 5*x + 7)^2 < 1
Detail solution
Given the inequality:
$$- \left(x - 3\right) \left(x + 2\right) + \left(\left(x^{2} + 5 x\right) + 7\right)^{2} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$- \left(x - 3\right) \left(x + 2\right) + \left(\left(x^{2} + 5 x\right) + 7\right)^{2} = 1$$
Solve:
$$x_{1} = -2$$
$$x_{2} = - \frac{8}{3} + \frac{2}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3}$$
$$x_{3} = - \frac{8}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
$$x_{4} = - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
Exclude the complex solutions:
$$x_{1} = -2$$
$$x_{2} = - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
This roots
$$x_{2} = - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
$$x_{1} = -2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(- \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right) + - \frac{1}{10}$$
=
$$- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
substitute to the expression
$$- \left(x - 3\right) \left(x + 2\right) + \left(\left(x^{2} + 5 x\right) + 7\right)^{2} < 1$$
$$- \left(\left(- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right) - 3\right) \left(\left(- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right) + 2\right) + \left(\left(5 \left(- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right) + \left(- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right)^{2}\right) + 7\right)^{2} < 1$$
but
Then
$$x < - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}} \wedge x < -2$$
$$- \left(x - 3\right) \left(x + 2\right) + \left(\left(x^{2} + 5 x\right) + 7\right)^{2} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$- \left(x - 3\right) \left(x + 2\right) + \left(\left(x^{2} + 5 x\right) + 7\right)^{2} = 1$$
Solve:
$$x_{1} = -2$$
$$x_{2} = - \frac{8}{3} + \frac{2}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3}$$
$$x_{3} = - \frac{8}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
$$x_{4} = - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
Exclude the complex solutions:
$$x_{1} = -2$$
$$x_{2} = - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
This roots
$$x_{2} = - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
$$x_{1} = -2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(- \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right) + - \frac{1}{10}$$
=
$$- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
substitute to the expression
$$- \left(x - 3\right) \left(x + 2\right) + \left(\left(x^{2} + 5 x\right) + 7\right)^{2} < 1$$
$$- \left(\left(- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right) - 3\right) \left(\left(- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right) + 2\right) + \left(\left(5 \left(- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right) + \left(- \frac{83}{30} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}\right)^{2}\right) + 7\right)^{2} < 1$$
2 / 2 \ | / _________________ \ _________________ | / _________________ \ / _________________ \ | | / _____ | / _____ | | / _____ | | / _____ | | | / 169 9*\/ 353 | / 169 9*\/ 353 | | / 169 9*\/ 353 | | / 169 9*\/ 353 | | | 3 / --- + --------- | 5*3 / --- + --------- | | 3 / --- + --------- | | 3 / --- + --------- | | 41 | 83 \/ 2 2 2 | \/ 2 2 10 | | 173 \/ 2 2 2 | | 23 \/ 2 2 2 | < 1 |- -- + |- -- - ---------------------- + ------------------------| - ------------------------ + ------------------------| - |- --- - ---------------------- + ------------------------|*|- -- - ---------------------- + ------------------------| | 6 | 30 3 _________________| 3 _________________| | 30 3 _________________| | 30 3 _________________| | | / _____ | / _____ | | / _____ | | / _____ | | | / 169 9*\/ 353 | / 169 9*\/ 353 | | / 169 9*\/ 353 | | / 169 9*\/ 353 | | | 3*3 / --- + --------- | 3*3 / --- + --------- | | 3*3 / --- + --------- | | 3*3 / --- + --------- | \ \ \/ 2 2 / \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
but
2 / 2 \ | / _________________ \ _________________ | / _________________ \ / _________________ \ | | / _____ | / _____ | | / _____ | | / _____ | | | / 169 9*\/ 353 | / 169 9*\/ 353 | | / 169 9*\/ 353 | | / 169 9*\/ 353 | | | 3 / --- + --------- | 5*3 / --- + --------- | | 3 / --- + --------- | | 3 / --- + --------- | | 41 | 83 \/ 2 2 2 | \/ 2 2 10 | | 173 \/ 2 2 2 | | 23 \/ 2 2 2 | > 1 |- -- + |- -- - ---------------------- + ------------------------| - ------------------------ + ------------------------| - |- --- - ---------------------- + ------------------------|*|- -- - ---------------------- + ------------------------| | 6 | 30 3 _________________| 3 _________________| | 30 3 _________________| | 30 3 _________________| | | / _____ | / _____ | | / _____ | | / _____ | | | / 169 9*\/ 353 | / 169 9*\/ 353 | | / 169 9*\/ 353 | | / 169 9*\/ 353 | | | 3*3 / --- + --------- | 3*3 / --- + --------- | | 3*3 / --- + --------- | | 3*3 / --- + --------- | \ \ \/ 2 2 / \/ 2 2 / \ \/ 2 2 / \ \/ 2 2 /
Then
$$x < - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{8}{3} - \frac{\sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}}{3} + \frac{2}{3 \sqrt[3]{\frac{169}{2} + \frac{9 \sqrt{353}}{2}}} \wedge x < -2$$
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x2 x1
Solving inequality on a graph
Rapid solution
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/ / 3 2 \ \ And\x < -2, CRootOf\x + 8*x + 22*x + 27, 0/ < x/
$$x < -2 \wedge \operatorname{CRootOf} {\left(x^{3} + 8 x^{2} + 22 x + 27, 0\right)} < x$$
(x < -2)∧(CRootOf(x^3 + 8*x^2 + 22*x + 27, 0) < x)
Rapid solution 2
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/ 3 2 \ (CRootOf\x + 8*x + 22*x + 27, 0/, -2)
$$x\ in\ \left(\operatorname{CRootOf} {\left(x^{3} + 8 x^{2} + 22 x + 27, 0\right)}, -2\right)$$
x in Interval.open(CRootOf(x^3 + 8*x^2 + 22*x + 27, 0), -2)