Mister Exam
(x+1)*(x-3)*(x+9)/((x-7)*(x-7))<0 inequation
The solution
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(x + 1)*(x - 3)*(x + 9)
----------------------- < 0
(x - 7)*(x - 7)
$$\frac{\left(x - 3\right) \left(x + 1\right) \left(x + 9\right)}{\left(x - 7\right) \left(x - 7\right)} < 0$$
(((x - 3)*(x + 1))*(x + 9))/(((x - 7)*(x - 7))) < 0
Detail solution
Given the inequality:
$$\frac{\left(x - 3\right) \left(x + 1\right) \left(x + 9\right)}{\left(x - 7\right) \left(x - 7\right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x - 3\right) \left(x + 1\right) \left(x + 9\right)}{\left(x - 7\right) \left(x - 7\right)} = 0$$
Solve:
$$x_{1} = -9$$
$$x_{2} = -1$$
$$x_{3} = 3$$
$$x_{1} = -9$$
$$x_{2} = -1$$
$$x_{3} = 3$$
This roots
$$x_{1} = -9$$
$$x_{2} = -1$$
$$x_{3} = 3$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-9 + - \frac{1}{10}$$
=
$$- \frac{91}{10}$$
substitute to the expression
$$\frac{\left(x - 3\right) \left(x + 1\right) \left(x + 9\right)}{\left(x - 7\right) \left(x - 7\right)} < 0$$
$$\frac{\left(- \frac{91}{10} - 3\right) \left(- \frac{91}{10} + 1\right) \left(- \frac{91}{10} + 9\right)}{\left(- \frac{91}{10} - 7\right) \left(- \frac{91}{10} - 7\right)} < 0$$
one of the solutions of our inequality is:
$$x < -9$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -9$$
$$x > -1 \wedge x < 3$$
$$\frac{\left(x - 3\right) \left(x + 1\right) \left(x + 9\right)}{\left(x - 7\right) \left(x - 7\right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x - 3\right) \left(x + 1\right) \left(x + 9\right)}{\left(x - 7\right) \left(x - 7\right)} = 0$$
Solve:
$$x_{1} = -9$$
$$x_{2} = -1$$
$$x_{3} = 3$$
$$x_{1} = -9$$
$$x_{2} = -1$$
$$x_{3} = 3$$
This roots
$$x_{1} = -9$$
$$x_{2} = -1$$
$$x_{3} = 3$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-9 + - \frac{1}{10}$$
=
$$- \frac{91}{10}$$
substitute to the expression
$$\frac{\left(x - 3\right) \left(x + 1\right) \left(x + 9\right)}{\left(x - 7\right) \left(x - 7\right)} < 0$$
$$\frac{\left(- \frac{91}{10} - 3\right) \left(- \frac{91}{10} + 1\right) \left(- \frac{91}{10} + 9\right)}{\left(- \frac{91}{10} - 7\right) \left(- \frac{91}{10} - 7\right)} < 0$$
-9801 ------ < 0 259210
one of the solutions of our inequality is:
$$x < -9$$
_____ _____
\ / \
-------ο-------ο-------ο-------
x1 x2 x3Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -9$$
$$x > -1 \wedge x < 3$$
Solving inequality on a graph
Rapid solution
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Or(And(-oo < x, x < -9), And(-1 < x, x < 3))
$$\left(-\infty < x \wedge x < -9\right) \vee \left(-1 < x \wedge x < 3\right)$$
((-oo < x)∧(x < -9))∨((-1 < x)∧(x < 3))
Rapid solution 2
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(-oo, -9) U (-1, 3)
$$x\ in\ \left(-\infty, -9\right) \cup \left(-1, 3\right)$$
x in Union(Interval.open(-oo, -9), Interval.open(-1, 3))