0,5*(4^x+6)>1 inequation

Given the inequality:
$$\frac{4^{x} + 6}{2} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{4^{x} + 6}{2} = 1$$
Solve:
Given the equation:
$$\frac{4^{x} + 6}{2} = 1$$
or
$$\frac{4^{x} + 6}{2} - 1 = 0$$
or
$$\frac{4^{x}}{2} = -2$$
or
$$4^{x} = -4$$
- this is the simplest exponential equation
Do replacement
$$v = 4^{x}$$
we get
$$v + 4 = 0$$
or
$$v + 4 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = -4$$
do backward replacement
$$4^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(4 \right)}}$$
$$x_{1} = -4$$
$$x_{1} = -4$$
This roots
$$x_{1} = -4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-4 + - \frac{1}{10}$$
=
$$- \frac{41}{10}$$
substitute to the expression
$$\frac{4^{x} + 6}{2} > 1$$
$$\frac{\frac{1}{4^{\frac{41}{10}}} + 6}{2} > 1$$

     4/5    
    2       
3 + ---- > 1
    1024    
    

the solution of our inequality is:
$$x < -4$$

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