Mister Exam

Lang:

log(3)(2x+1)<2 inequation

You have entered [src]

log(3)*(2*x + 1) < 2

\left(2 x + 1\right) \log{\left(3 \right)} < 2

(2*x + 1)*log(3) < 2

Detail solution

Given the inequality:

\left(2 x + 1\right) \log{\left(3 \right)} < 2

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

\left(2 x + 1\right) \log{\left(3 \right)} = 2

Solve:
Given the equation:

log(3)*(2*x+1) = 2

Expand expressions:

2*x*log(3) + log(3) = 2

Reducing, you get:

-2 + 2*x*log(3) + log(3) = 0

Expand brackets in the left part

-2 + 2*x*log3 + log3 = 0

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

2 x \log{\left(3 \right)} + \log{\left(3 \right)} = 2

Divide both parts of the equation by (2*x*log(3) + log(3))/x

x = 2 / ((2*x*log(3) + log(3))/x)

We get the answer: x = -1/2 + 1/log(3)

x_{1} = - \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}

x_{1} = - \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}

This roots

x_{1} = - \frac{1}{2} + \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} < x_{1}

For example, let's take the point

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

- \frac{1}{10} + \left(- \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}\right)

- \frac{3}{5} + \frac{1}{\log{\left(3 \right)}}

substitute to the expression

\left(2 x + 1\right) \log{\left(3 \right)} < 2

\left(2 \left(- \frac{3}{5} + \frac{1}{\log{\left(3 \right)}}\right) + 1\right) \log{\left(3 \right)} < 2

/  1     2   \           
|- - + ------|*log(3) < 2
\  5   log(3)/

the solution of our inequality is:

x < - \frac{1}{2} + \frac{1}{\log{\left(3 \right)}}

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

Solving inequality on a graph

Rapid solution [src]

   /             2 - log(3)\
And|-oo < x, x < ----------|
   \              2*log(3) /

-\infty < x \wedge x < \frac{2 - \log{\left(3 \right)}}{2 \log{\left(3 \right)}}

(-oo < x)∧(x < (2 - log(3))/(2*log(3)))

Rapid solution 2 [src]

      2 - log(3) 
(-oo, ----------)
       2*log(3)

x\ in\ \left(-\infty, \frac{2 - \log{\left(3 \right)}}{2 \log{\left(3 \right)}}\right)

x in Interval.open(-oo, (2 - log(3))/(2*log(3)))

log(3)*(2*x+1)<2 inequation