Mister Exam
12*x^4+12*x^3-12*x^2-1<=0 inequation
The solution
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4 3 2 12*x + 12*x - 12*x - 1 <= 0
$$\left(- 12 x^{2} + \left(12 x^{4} + 12 x^{3}\right)\right) - 1 \leq 0$$
-12*x^2 + 12*x^4 + 12*x^3 - 1 <= 0
Detail solution
Given the inequality:
$$\left(- 12 x^{2} + \left(12 x^{4} + 12 x^{3}\right)\right) - 1 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 12 x^{2} + \left(12 x^{4} + 12 x^{3}\right)\right) - 1 = 0$$
Solve:
$$x_{1} = - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2}$$
$$x_{2} = - \frac{1}{4} + \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} - \frac{\sqrt{- \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}} + \frac{\sqrt[3]{82}}{6} + \frac{11}{6}}}{2}$$
$$x_{3} = - \frac{1}{4} + \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{- \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}} + \frac{\sqrt[3]{82}}{6} + \frac{11}{6}}}{2}$$
$$x_{4} = - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
Exclude the complex solutions:
$$x_{1} = - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2}$$
$$x_{2} = - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
This roots
$$x_{2} = - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
$$x_{1} = - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{7}{20} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
substitute to the expression
$$\left(- 12 x^{2} + \left(12 x^{4} + 12 x^{3}\right)\right) - 1 \leq 0$$
$$-1 + \left(- 12 \left(- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{7}{20} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}\right)^{2} + \left(12 \left(- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{7}{20} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}\right)^{3} + 12 \left(- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{7}{20} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}\right)^{4}\right)\right) \leq 0$$
but
Then
$$x \leq - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} \wedge x \leq - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2}$$
$$\left(- 12 x^{2} + \left(12 x^{4} + 12 x^{3}\right)\right) - 1 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 12 x^{2} + \left(12 x^{4} + 12 x^{3}\right)\right) - 1 = 0$$
Solve:
$$x_{1} = - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2}$$
$$x_{2} = - \frac{1}{4} + \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} - \frac{\sqrt{- \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}} + \frac{\sqrt[3]{82}}{6} + \frac{11}{6}}}{2}$$
$$x_{3} = - \frac{1}{4} + \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{- \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}} + \frac{\sqrt[3]{82}}{6} + \frac{11}{6}}}{2}$$
$$x_{4} = - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
Exclude the complex solutions:
$$x_{1} = - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2}$$
$$x_{2} = - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
This roots
$$x_{2} = - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
$$x_{1} = - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{7}{20} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
substitute to the expression
$$\left(- 12 x^{2} + \left(12 x^{4} + 12 x^{3}\right)\right) - 1 \leq 0$$
$$-1 + \left(- 12 \left(- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{7}{20} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}\right)^{2} + \left(12 \left(- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{7}{20} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}\right)^{3} + 12 \left(- \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{7}{20} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}\right)^{4}\right)\right) \leq 0$$
2 3 4
/ ____________________________________\ / ____________________________________\ / ____________________________________\
| / 3 ____ | | / 3 ____ | | / 3 ____ |
| / 11 \/ 82 5 | | / 11 \/ 82 5 | | / 11 \/ 82 5 |
| / -- + ------ + -------------------- | | / -- + ------ + -------------------- | | / -- + ------ + -------------------- |
| _____________ / 6 6 _____________ | | _____________ / 6 6 _____________ | | _____________ / 6 6 _____________ |
| / 3 ____ / / 3 ____ | | / 3 ____ / / 3 ____ | | / 3 ____ / / 3 ____ | <= 0
| / 11 \/ 82 / / 11 \/ 82 | | / 11 \/ 82 / / 11 \/ 82 | | / 11 \/ 82 / / 11 \/ 82 |
| / -- - ------ / 4* / -- - ------ | | / -- - ------ / 4* / -- - ------ | | / -- - ------ / 4* / -- - ------ |
| 7 \/ 12 6 \/ \/ 12 6 | | 7 \/ 12 6 \/ \/ 12 6 | | 7 \/ 12 6 \/ \/ 12 6 |
-1 - 12*|- -- - ------------------ - ---------------------------------------------| + 12*|- -- - ------------------ - ---------------------------------------------| + 12*|- -- - ------------------ - ---------------------------------------------|
\ 20 2 2 / \ 20 2 2 / \ 20 2 2 /
but
2 3 4
/ ____________________________________\ / ____________________________________\ / ____________________________________\
| / 3 ____ | | / 3 ____ | | / 3 ____ |
| / 11 \/ 82 5 | | / 11 \/ 82 5 | | / 11 \/ 82 5 |
| / -- + ------ + -------------------- | | / -- + ------ + -------------------- | | / -- + ------ + -------------------- |
| _____________ / 6 6 _____________ | | _____________ / 6 6 _____________ | | _____________ / 6 6 _____________ |
| / 3 ____ / / 3 ____ | | / 3 ____ / / 3 ____ | | / 3 ____ / / 3 ____ | >= 0
| / 11 \/ 82 / / 11 \/ 82 | | / 11 \/ 82 / / 11 \/ 82 | | / 11 \/ 82 / / 11 \/ 82 |
| / -- - ------ / 4* / -- - ------ | | / -- - ------ / 4* / -- - ------ | | / -- - ------ / 4* / -- - ------ |
| 7 \/ 12 6 \/ \/ 12 6 | | 7 \/ 12 6 \/ \/ 12 6 | | 7 \/ 12 6 \/ \/ 12 6 |
-1 - 12*|- -- - ------------------ - ---------------------------------------------| + 12*|- -- - ------------------ - ---------------------------------------------| + 12*|- -- - ------------------ - ---------------------------------------------|
\ 20 2 2 / \ 20 2 2 / \ 20 2 2 /
Then
$$x \leq - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2} - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} \wedge x \leq - \frac{1}{4} - \frac{\sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}{2} + \frac{\sqrt{\frac{\sqrt[3]{82}}{6} + \frac{11}{6} + \frac{5}{4 \sqrt{\frac{11}{12} - \frac{\sqrt[3]{82}}{6}}}}}{2}$$
_____
/ \
-------•-------•-------
x2 x1
Solving inequality on a graph
Rapid solution
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/ / 4 3 2 \ / 4 3 2 \ \ And\x <= CRootOf\12*x + 12*x - 12*x - 1, 1/, CRootOf\12*x + 12*x - 12*x - 1, 0/ <= x/
$$x \leq \operatorname{CRootOf} {\left(12 x^{4} + 12 x^{3} - 12 x^{2} - 1, 1\right)} \wedge \operatorname{CRootOf} {\left(12 x^{4} + 12 x^{3} - 12 x^{2} - 1, 0\right)} \leq x$$
(x <= CRootOf(12*x^4 + 12*x^3 - 12*x^2 - 1, 1))∧(CRootOf(12*x^4 + 12*x^3 - 12*x^2 - 1, 0) <= x)
Rapid solution 2
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/ 4 3 2 \ / 4 3 2 \ [CRootOf\12*x + 12*x - 12*x - 1, 0/, CRootOf\12*x + 12*x - 12*x - 1, 1/]
$$x\ in\ \left[\operatorname{CRootOf} {\left(12 x^{4} + 12 x^{3} - 12 x^{2} - 1, 0\right)}, \operatorname{CRootOf} {\left(12 x^{4} + 12 x^{3} - 12 x^{2} - 1, 1\right)}\right]$$
x in Interval(CRootOf(12*x^4 + 12*x^3 - 12*x^2 - 1, 0), CRootOf(12*x^4 + 12*x^3 - 12*x^2 - 1, 1))