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(x-3)*(x-5)>0 inequation

A inequation with variable

The solution

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(x - 3)*(x - 5) > 0
$$\left(x - 5\right) \left(x - 3\right) > 0$$
(x - 5)*(x - 3) > 0
Detail solution
Given the inequality:
$$\left(x - 5\right) \left(x - 3\right) > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x - 5\right) \left(x - 3\right) = 0$$
Solve:
Expand the expression in the equation
$$\left(x - 5\right) \left(x - 3\right) = 0$$
We get the quadratic equation
$$x^{2} - 8 x + 15 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -8$$
$$c = 15$$
, then
D = b^2 - 4 * a * c = 

(-8)^2 - 4 * (1) * (15) = 4

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = 5$$
$$x_{2} = 3$$
$$x_{1} = 5$$
$$x_{2} = 3$$
$$x_{1} = 5$$
$$x_{2} = 3$$
This roots
$$x_{2} = 3$$
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3$$
=
$$\frac{29}{10}$$
substitute to the expression
$$\left(x - 5\right) \left(x - 3\right) > 0$$
$$\left(-5 + \frac{29}{10}\right) \left(-3 + \frac{29}{10}\right) > 0$$
 21    
--- > 0
100    

one of the solutions of our inequality is:
$$x < 3$$
 _____           _____          
      \         /
-------ο-------ο-------
       x2      x1

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 3$$
$$x > 5$$
Solving inequality on a graph
Rapid solution 2 [src]
(-oo, 3) U (5, oo)
$$x\ in\ \left(-\infty, 3\right) \cup \left(5, \infty\right)$$
x in Union(Interval.open(-oo, 3), Interval.open(5, oo))
Rapid solution [src]
Or(And(-oo < x, x < 3), And(5 < x, x < oo))
$$\left(-\infty < x \wedge x < 3\right) \vee \left(5 < x \wedge x < \infty\right)$$
((-oo < x)∧(x < 3))∨((5 < x)∧(x < oo))