25x²<49 inequation

Given the inequality:
$$25 x^{2} < 49$$
To solve this inequality, we must first solve the corresponding equation:
$$25 x^{2} = 49$$
Solve:
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$25 x^{2} = 49$$
to
$$25 x^{2} - 49 = 0$$
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
$$a = 25$$
$$b = 0$$
$$c = -49$$
, then

D = b^2 - 4 * a * c = 

(0)^2 - 4 * (25) * (-49) = 4900

Because D > 0, then the equation has two roots.

x1 = (-b + sqrt(D)) / (2*a)

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

or
$$x_{1} = \frac{7}{5}$$
$$x_{2} = - \frac{7}{5}$$
$$x_{1} = \frac{7}{5}$$
$$x_{2} = - \frac{7}{5}$$
$$x_{1} = \frac{7}{5}$$
$$x_{2} = - \frac{7}{5}$$
This roots
$$x_{2} = - \frac{7}{5}$$
$$x_{1} = \frac{7}{5}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{7}{5} + - \frac{1}{10}$$
=
$$- \frac{3}{2}$$
substitute to the expression
$$25 x^{2} < 49$$
$$25 \left(- \frac{3}{2}\right)^{2} < 49$$

225/4 < 49

but

225/4 > 49

Then
$$x < - \frac{7}{5}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{7}{5} \wedge x < \frac{7}{5}$$

         _____  
        /     \  
-------ο-------ο-------
       x2      x1