9^x-3^x+4<82 inequation

Given the inequality:
$$- 3^{x} + 9^{x} + 4 < 82$$
To solve this inequality, we must first solve the corresponding equation:
$$- 3^{x} + 9^{x} + 4 = 82$$
Solve:
Given the equation:
$$- 3^{x} + 9^{x} + 4 = 82$$
or
$$\left(- 3^{x} + 9^{x} + 4\right) - 82 = 0$$
Do replacement
$$v = 3^{x}$$
we get
$$v^{2} - v - 78 = 0$$
or
$$v^{2} - v - 78 = 0$$
This equation is of the form

a*v^2 + b*v + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = -78$$
, then

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

(-1)^2 - 4 * (1) * (-78) = 313

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

v1 = (-b + sqrt(D)) / (2*a)

v2 = (-b - sqrt(D)) / (2*a)

or
$$v_{1} = \frac{1}{2} + \frac{\sqrt{313}}{2}$$
Simplify
$$v_{2} = \frac{1}{2} - \frac{\sqrt{313}}{2}$$
Simplify
do backward replacement
$$3^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(3 \right)}}$$
$$x_{1} = \frac{1}{2} - \frac{\sqrt{313}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{313}}{2}$$
$$x_{1} = \frac{1}{2} - \frac{\sqrt{313}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{313}}{2}$$
This roots
$$x_{1} = \frac{1}{2} - \frac{\sqrt{313}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{313}}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{1}{2} - \frac{\sqrt{313}}{2}\right) - \frac{1}{10}$$
=
$$\frac{2}{5} - \frac{\sqrt{313}}{2}$$
substitute to the expression
$$- 3^{x} + 9^{x} + 4 < 82$$
$$- 3^{\frac{2}{5} - \frac{\sqrt{313}}{2}} + 9^{\frac{2}{5} - \frac{\sqrt{313}}{2}} + 4 < 82$$

           _____          _____     
     2   \/ 313     2   \/ 313      
     - - -------    - - ------- < 82
     5      2       5      2        
4 + 9            - 3                

one of the solutions of our inequality is:
$$x < \frac{1}{2} - \frac{\sqrt{313}}{2}$$

 _____           _____          
      \         /
-------ο-------ο-------
       x_1      x_2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{1}{2} - \frac{\sqrt{313}}{2}$$
$$x > \frac{1}{2} + \frac{\sqrt{313}}{2}$$