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log(1/3)x≤−1. inequation

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log(1/3)*x <= -1

x \log{\left(\frac{1}{3} \right)} \leq -1

x*log(1/3) <= -1

Detail solution

Given the inequality:

x \log{\left(\frac{1}{3} \right)} \leq -1

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

x \log{\left(\frac{1}{3} \right)} = -1

Solve:
Given the linear equation:

log(1/3)*x = -1

Expand brackets in the left part

log1/3x = -1

Divide both parts of the equation by -log(3)

x = -1 / (-log(3))

x_{1} = \frac{1}{\log{\left(3 \right)}}

x_{1} = \frac{1}{\log{\left(3 \right)}}

This roots

x_{1} = \frac{1}{\log{\left(3 \right)}}

is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:

x_{0} \leq x_{1}

For example, let's take the point

x_{0} = x_{1} - \frac{1}{10}

- \frac{1}{10} + \frac{1}{\log{\left(3 \right)}}

- \frac{1}{10} + \frac{1}{\log{\left(3 \right)}}

substitute to the expression

x \log{\left(\frac{1}{3} \right)} \leq -1

\left(- \frac{1}{10} + \frac{1}{\log{\left(3 \right)}}\right) \log{\left(\frac{1}{3} \right)} \leq -1

 /  1      1   \             
-|- -- + ------|*log(3) <= -1
 \  10   log(3)/

but

 /  1      1   \             
-|- -- + ------|*log(3) >= -1
 \  10   log(3)/

Then

x \leq \frac{1}{\log{\left(3 \right)}}

no execute
the solution of our inequality is:

x \geq \frac{1}{\log{\left(3 \right)}}

         _____  
        /
-------•-------
       x1

Solving inequality on a graph

Rapid solution 2 [src]

   1        
[------, oo)
 log(3)

x\ in\ \left[\frac{1}{\log{\left(3 \right)}}, \infty\right)

x in Interval(1/log(3), oo)

Rapid solution [src]

   /  1                \
And|------ <= x, x < oo|
   \log(3)             /

\frac{1}{\log{\left(3 \right)}} \leq x \wedge x < \infty

(x < oo)∧(1/log(3) <= x)