Mister Exam
x/(x+3)-(-1+3/x)+13/(x^2+2*x-3)<0 inequation
The solution
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x 3 13
----- + 1 - - + ------------ < 0
x + 3 x 2
x + 2*x - 3
$$\left(\frac{x}{x + 3} + \left(1 - \frac{3}{x}\right)\right) + \frac{13}{\left(x^{2} + 2 x\right) - 3} < 0$$
x/(x + 3) + 1 - 3/x + 13/(x^2 + 2*x - 3) < 0
Detail solution
Given the inequality:
$$\left(\frac{x}{x + 3} + \left(1 - \frac{3}{x}\right)\right) + \frac{13}{\left(x^{2} + 2 x\right) - 3} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{x}{x + 3} + \left(1 - \frac{3}{x}\right)\right) + \frac{13}{\left(x^{2} + 2 x\right) - 3} = 0$$
Solve:
$$x_{1} = \frac{1}{3} + \frac{5}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3}$$
$$x_{2} = \frac{1}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}$$
$$x_{3} = - \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}$$
Exclude the complex solutions:
$$x_{1} = - \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}$$
This roots
$$x_{1} = - \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}$$
substitute to the expression
$$\left(\frac{x}{x + 3} + \left(1 - \frac{3}{x}\right)\right) + \frac{13}{\left(x^{2} + 2 x\right) - 3} < 0$$
$$\frac{13}{-3 + \left(2 \left(- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}\right) + \left(- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}\right)^{2}\right)} + \left(\frac{- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}}{\left(- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}\right) + 3} + \left(1 - \frac{3}{- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}}\right)\right) < 0$$
the solution of our inequality is:
$$x < - \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}$$
$$\left(\frac{x}{x + 3} + \left(1 - \frac{3}{x}\right)\right) + \frac{13}{\left(x^{2} + 2 x\right) - 3} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{x}{x + 3} + \left(1 - \frac{3}{x}\right)\right) + \frac{13}{\left(x^{2} + 2 x\right) - 3} = 0$$
Solve:
$$x_{1} = \frac{1}{3} + \frac{5}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3}$$
$$x_{2} = \frac{1}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}$$
$$x_{3} = - \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}$$
Exclude the complex solutions:
$$x_{1} = - \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}$$
This roots
$$x_{1} = - \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}$$
substitute to the expression
$$\left(\frac{x}{x + 3} + \left(1 - \frac{3}{x}\right)\right) + \frac{13}{\left(x^{2} + 2 x\right) - 3} < 0$$
$$\frac{13}{-3 + \left(2 \left(- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}\right) + \left(- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}\right)^{2}\right)} + \left(\frac{- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}}{\left(- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}\right) + 3} + \left(1 - \frac{3}{- \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{7}{30} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}}\right)\right) < 0$$
__________________
/ _____
/ 275 15*\/ 345
3 / --- + ----------
7 \/ 4 4 5
-- - ----------------------- + -------------------------
30 3 __________________
/ _____
/ 275 15*\/ 345
3*3 / --- + ----------
3 13 \/ 4 4
1 - -------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------
__________________ 2 __________________ < 0
/ _____ / __________________ \ __________________ / _____
/ 275 15*\/ 345 | / _____ | / _____ / 275 15*\/ 345
3 / --- + ---------- | / 275 15*\/ 345 | / 275 15*\/ 345 3 / --- + ----------
7 \/ 4 4 5 | 3 / --- + ---------- | 2*3 / --- + ---------- 97 \/ 4 4 5
-- - ----------------------- + ------------------------- 38 |7 \/ 4 4 5 | \/ 4 4 10 -- - ----------------------- + -------------------------
30 3 __________________ - -- + |-- - ----------------------- + -------------------------| - ------------------------- + ------------------------- 30 3 __________________
/ _____ 15 |30 3 __________________| 3 __________________ / _____
/ 275 15*\/ 345 | / _____ | / _____ / 275 15*\/ 345
3*3 / --- + ---------- | / 275 15*\/ 345 | / 275 15*\/ 345 3*3 / --- + ----------
\/ 4 4 | 3*3 / --- + ---------- | 3*3 / --- + ---------- \/ 4 4
\ \/ 4 4 / \/ 4 4
the solution of our inequality is:
$$x < - \frac{\sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}}{3} + \frac{5}{3 \sqrt[3]{\frac{275}{4} + \frac{15 \sqrt{345}}{4}}} + \frac{1}{3}$$
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x1
Solving inequality on a graph
Rapid solution
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/ / / 3 2 \\ \ Or\And\-3 < x, x < CRootOf\2*x - 2*x + 4*x + 9, 0//, And(0 < x, x < 1)/
$$\left(-3 < x \wedge x < \operatorname{CRootOf} {\left(2 x^{3} - 2 x^{2} + 4 x + 9, 0\right)}\right) \vee \left(0 < x \wedge x < 1\right)$$
((0 < x)∧(x < 1))∨((-3 < x)∧(x < CRootOf(2*x^3 - 2*x^2 + 4*x + 9, 0)))
Rapid solution 2
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/ 3 2 \ (-3, CRootOf\2*x - 2*x + 4*x + 9, 0/) U (0, 1)
$$x\ in\ \left(-3, \operatorname{CRootOf} {\left(2 x^{3} - 2 x^{2} + 4 x + 9, 0\right)}\right) \cup \left(0, 1\right)$$
x in Union(Interval.open(-3, CRootOf(2*x^3 - 2*x^2 + 4*x + 9, 0)), Interval.open(0, 1))