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(1/2)^(x+2)>4 inequation

You have entered [src]

 -2 - x    
2       > 4

\left(\frac{1}{2}\right)^{x + 2} > 4

(1/2)^(x + 2) > 4

Detail solution

Given the inequality:

\left(\frac{1}{2}\right)^{x + 2} > 4

To solve this inequality, we must first solve the corresponding equation:

\left(\frac{1}{2}\right)^{x + 2} = 4

Solve:
Given the equation:

\left(\frac{1}{2}\right)^{x + 2} = 4

\left(\frac{1}{2}\right)^{x + 2} - 4 = 0

\frac{2^{- x}}{4} = 4

\left(\frac{1}{2}\right)^{x} = 16

- this is the simplest exponential equation
Do replacement

v = \left(\frac{1}{2}\right)^{x}

we get

v - 16 = 0

v - 16 = 0

Move free summands (without v)
from left part to right part, we given:

v = 16

do backward replacement

\left(\frac{1}{2}\right)^{x} = v

x = - \frac{\log{\left(v \right)}}{\log{\left(2 \right)}}

x_{1} = 16

x_{1} = 16

This roots

x_{1} = 16

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} + 16

\frac{159}{10}

substitute to the expression

\left(\frac{1}{2}\right)^{x + 2} > 4

\left(\frac{1}{2}\right)^{2 + \frac{159}{10}} > 4

10___     
\/ 2      
------ > 4
262144

Then

x < 16

no execute
the solution of our inequality is:

x > 16

         _____  
        /
-------ο-------
       x1

Solving inequality on a graph

Rapid solution [src]

         log(4)
x < -2 - ------
         log(2)

x < -2 - \frac{\log{\left(4 \right)}}{\log{\left(2 \right)}}

x < -2 - log(4)/log(2)

Rapid solution 2 [src]

           log(4) 
(-oo, -2 - ------)
           log(2)

x\ in\ \left(-\infty, -2 - \frac{\log{\left(4 \right)}}{\log{\left(2 \right)}}\right)

x in Interval.open(-oo, -2 - log(4)/log(2))