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

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 -1 - 2*x    
2         < 4

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

(1/2)^(2*x + 1) < 4

Detail solution

Given the inequality:

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

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

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

Solve:
Given the equation:

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

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

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

\left(\frac{1}{4}\right)^{x} = 8

- this is the simplest exponential equation
Do replacement

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

we get

v - 8 = 0

v - 8 = 0

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

v = 8

do backward replacement

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

x = - \frac{\log{\left(v \right)}}{\log{\left(4 \right)}}

x_{1} = 8

x_{1} = 8

This roots

x_{1} = 8

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} + 8

\frac{79}{10}

substitute to the expression

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

\left(\frac{1}{2}\right)^{1 + \frac{2 \cdot 79}{10}} < 4

5 ___     
\/ 2      
------ < 4
131072

the solution of our inequality is:

x < 8

 _____          
      \    
-------ο-------
       x1

Solving inequality on a graph

Rapid solution [src]

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

- \frac{\log{\left(4 \right)}}{2 \log{\left(2 \right)}} - \frac{1}{2} < x

-log(4)/(2*log(2)) - 1/2 < x

Rapid solution 2 [src]

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

x\ in\ \left(- \frac{\log{\left(4 \right)}}{2 \log{\left(2 \right)}} - \frac{1}{2}, \infty\right)

x in Interval.open(-log(4)/(2*log(2)) - 1/2, oo)