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5^(1-x)>0,2 inequation

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 1 - x      
5      > 1/5

5^{1 - x} > \frac{1}{5}

5^(1 - x) > 1/5

Detail solution

Given the inequality:

5^{1 - x} > \frac{1}{5}

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

5^{1 - x} = \frac{1}{5}

Solve:
Given the equation:

5^{1 - x} = \frac{1}{5}

5^{1 - x} - \frac{1}{5} = 0

5 \cdot 5^{- x} = \frac{1}{5}

\left(\frac{1}{5}\right)^{x} = \frac{1}{25}

- this is the simplest exponential equation
Do replacement

v = \left(\frac{1}{5}\right)^{x}

we get

v - \frac{1}{25} = 0

v - \frac{1}{25} = 0

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

v = \frac{1}{25}

do backward replacement

\left(\frac{1}{5}\right)^{x} = v

x = - \frac{\log{\left(v \right)}}{\log{\left(5 \right)}}

x_{1} = \frac{1}{25}

x_{1} = \frac{1}{25}

This roots

x_{1} = \frac{1}{25}

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} + \frac{1}{25}

- \frac{3}{50}

substitute to the expression

5^{1 - x} > \frac{1}{5}

5^{1 - - \frac{3}{50}} > \frac{1}{5}

   3/50      
5*5     > 1/5

the solution of our inequality is:

x < \frac{1}{25}

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Solving inequality on a graph

Rapid solution [src]

x < 2

x < 2

x < 2

Rapid solution 2 [src]

(-oo, 2)

x\ in\ \left(-\infty, 2\right)

x in Interval.open(-oo, 2)