Mister Exam
(x+2)*(x-3)*(x-4)/(x-2)^2>0 inequation
The solution
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[src]
(x + 2)*(x - 3)*(x - 4)
----------------------- > 0
2
(x - 2)
$$\frac{\left(x - 3\right) \left(x + 2\right) \left(x - 4\right)}{\left(x - 2\right)^{2}} > 0$$
(((x - 3)*(x + 2))*(x - 4))/(x - 2)^2 > 0
Detail solution
Given the inequality:
$$\frac{\left(x - 3\right) \left(x + 2\right) \left(x - 4\right)}{\left(x - 2\right)^{2}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x - 3\right) \left(x + 2\right) \left(x - 4\right)}{\left(x - 2\right)^{2}} = 0$$
Solve:
$$x_{1} = -2$$
$$x_{2} = 3$$
$$x_{3} = 4$$
$$x_{1} = -2$$
$$x_{2} = 3$$
$$x_{3} = 4$$
This roots
$$x_{1} = -2$$
$$x_{2} = 3$$
$$x_{3} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\frac{\left(x - 3\right) \left(x + 2\right) \left(x - 4\right)}{\left(x - 2\right)^{2}} > 0$$
$$\frac{\left(-3 + - \frac{21}{10}\right) \left(- \frac{21}{10} + 2\right) \left(-4 + - \frac{21}{10}\right)}{\left(- \frac{21}{10} - 2\right)^{2}} > 0$$
Then
$$x < -2$$
no execute
one of the solutions of our inequality is:
$$x > -2 \wedge x < 3$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > -2 \wedge x < 3$$
$$x > 4$$
$$\frac{\left(x - 3\right) \left(x + 2\right) \left(x - 4\right)}{\left(x - 2\right)^{2}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x - 3\right) \left(x + 2\right) \left(x - 4\right)}{\left(x - 2\right)^{2}} = 0$$
Solve:
$$x_{1} = -2$$
$$x_{2} = 3$$
$$x_{3} = 4$$
$$x_{1} = -2$$
$$x_{2} = 3$$
$$x_{3} = 4$$
This roots
$$x_{1} = -2$$
$$x_{2} = 3$$
$$x_{3} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\frac{\left(x - 3\right) \left(x + 2\right) \left(x - 4\right)}{\left(x - 2\right)^{2}} > 0$$
$$\frac{\left(-3 + - \frac{21}{10}\right) \left(- \frac{21}{10} + 2\right) \left(-4 + - \frac{21}{10}\right)}{\left(- \frac{21}{10} - 2\right)^{2}} > 0$$
-3111 ------ > 0 16810
Then
$$x < -2$$
no execute
one of the solutions of our inequality is:
$$x > -2 \wedge x < 3$$
_____ _____
/ \ /
-------ο-------ο-------ο-------
x1 x2 x3Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > -2 \wedge x < 3$$
$$x > 4$$
Solving inequality on a graph
Rapid solution
[src]
Or(And(-2 < x, x < 2), And(2 < x, x < 3), And(4 < x, x < oo))
$$\left(-2 < x \wedge x < 2\right) \vee \left(2 < x \wedge x < 3\right) \vee \left(4 < x \wedge x < \infty\right)$$
((-2 < x)∧(x < 2))∨((2 < x)∧(x < 3))∨((4 < x)∧(x < oo))
Rapid solution 2
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(-2, 2) U (2, 3) U (4, oo)
$$x\ in\ \left(-2, 2\right) \cup \left(2, 3\right) \cup \left(4, \infty\right)$$
x in Union(Interval.open(-2, 2), Interval.open(2, 3), Interval.open(4, oo))