Given the inequality:
$$\left(5 - x\right)^{2} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(5 - x\right)^{2} = 0$$
Solve:
Expand the expression in the equation
$$\left(5 - x\right)^{2} = 0$$
We get the quadratic equation
$$x^{2} - 10 x + 25 = 0$$
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
$$a = 1$$
$$b = -10$$
$$c = 25$$
, then
D = b^2 - 4 * a * c =
(-10)^2 - 4 * (1) * (25) = 0
Because D = 0, then the equation has one root.
x = -b/2a = --10/2/(1)
$$x_{1} = 5$$
$$x_{1} = 5$$
$$x_{1} = 5$$
This roots
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5$$
=
$$\frac{49}{10}$$
substitute to the expression
$$\left(5 - x\right)^{2} > 0$$
$$\left(5 - \frac{49}{10}\right)^{2} > 0$$
1/100 > 0
the solution of our inequality is:
$$x < 5$$
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