Mister Exam
100*1/(x-5)*(x-10)>0 inequation
The solution
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100 -----*(x - 10) > 0 x - 5
$$\left(x - 10\right) \frac{100}{x - 5} > 0$$
(x - 10)*(100/(x - 5)) > 0
Detail solution
Given the inequality:
$$\left(x - 10\right) \frac{100}{x - 5} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x - 10\right) \frac{100}{x - 5} = 0$$
Solve:
Given the equation:
$$\left(x - 10\right) \frac{100}{x - 5} = 0$$
Multiply the equation sides by the denominator -5 + x
we get:
$$100 x - 1000 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$100 x = 1000$$
Divide both parts of the equation by 100
$$x_{1} = 10$$
$$x_{1} = 10$$
This roots
$$x_{1} = 10$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 10$$
=
$$\frac{99}{10}$$
substitute to the expression
$$\left(x - 10\right) \frac{100}{x - 5} > 0$$
$$\left(-10 + \frac{99}{10}\right) \frac{100}{-5 + \frac{99}{10}} > 0$$
Then
$$x < 10$$
no execute
the solution of our inequality is:
$$x > 10$$
$$\left(x - 10\right) \frac{100}{x - 5} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x - 10\right) \frac{100}{x - 5} = 0$$
Solve:
Given the equation:
$$\left(x - 10\right) \frac{100}{x - 5} = 0$$
Multiply the equation sides by the denominator -5 + x
we get:
$$100 x - 1000 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$100 x = 1000$$
Divide both parts of the equation by 100
x = 1000 / (100)
$$x_{1} = 10$$
$$x_{1} = 10$$
This roots
$$x_{1} = 10$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 10$$
=
$$\frac{99}{10}$$
substitute to the expression
$$\left(x - 10\right) \frac{100}{x - 5} > 0$$
$$\left(-10 + \frac{99}{10}\right) \frac{100}{-5 + \frac{99}{10}} > 0$$
-100 ----- > 0 49
Then
$$x < 10$$
no execute
the solution of our inequality is:
$$x > 10$$
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Rapid solution
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Or(And(-oo < x, x < 5), And(10 < x, x < oo))
$$\left(-\infty < x \wedge x < 5\right) \vee \left(10 < x \wedge x < \infty\right)$$
((-oo < x)∧(x < 5))∨((10 < x)∧(x < oo))
Rapid solution 2
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(-oo, 5) U (10, oo)
$$x\ in\ \left(-\infty, 5\right) \cup \left(10, \infty\right)$$
x in Union(Interval.open(-oo, 5), Interval.open(10, oo))