log(6)x>1 inequation

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log(6)*x > 1

x \log{\left(6 \right)} > 1

x*log(6) > 1

Detail solution

Given the inequality:

x \log{\left(6 \right)} > 1

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

x \log{\left(6 \right)} = 1

Solve:
Given the linear equation:

log(6)*x = 1

Expand brackets in the left part

log6x = 1

Divide both parts of the equation by log(6)

x = 1 / (log(6))

x_{1} = \frac{1}{\log{\left(6 \right)}}

x_{1} = \frac{1}{\log{\left(6 \right)}}

This roots

x_{1} = \frac{1}{\log{\left(6 \right)}}

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}{\log{\left(6 \right)}}

=

- \frac{1}{10} + \frac{1}{\log{\left(6 \right)}}

substitute to the expression

x \log{\left(6 \right)} > 1

\left(- \frac{1}{10} + \frac{1}{\log{\left(6 \right)}}\right) \log{\left(6 \right)} > 1

/  1      1   \           
|- -- + ------|*log(6) > 1
\  10   log(6)/

Then

x < \frac{1}{\log{\left(6 \right)}}

no execute
the solution of our inequality is:

x > \frac{1}{\log{\left(6 \right)}}

         _____  
        /
-------ο-------
       x_1

Solving inequality on a graph

Rapid solution 2 [src]

   1        
(------, oo)
 log(6)

x\ in\ \left(\frac{1}{\log{\left(6 \right)}}, \infty\right)

x in Interval.open(1/log(6), oo)

Rapid solution [src]

   /          1       \
And|x < oo, ------ < x|
   \        log(6)    /

x < \infty \wedge \frac{1}{\log{\left(6 \right)}} < x

(x < oo)∧(1/log(6) < x)

The graph