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  • Similar expressions

  • arctan(2^x)<1
Solving inequalities/ atan(2^x)<1

atan(2^x)<1 inequation

A inequation with variable

The solution

You have entered [src] 
    / x\    
atan\2 / < 1
$$\operatorname{atan}{\left(2^{x} \right)} < 1$$ 
atan(2^x) < 1
Detail solution
Given the inequality:
$$\operatorname{atan}{\left(2^{x} \right)} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\operatorname{atan}{\left(2^{x} \right)} = 1$$
Solve:
$$x_{1} = \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}$$
$$x_{1} = \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}$$
This roots
$$x_{1} = \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}$$
=
$$- \frac{1}{10} + \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}$$
substitute to the expression
$$\operatorname{atan}{\left(2^{x} \right)} < 1$$
$$\operatorname{atan}{\left(2^{- \frac{1}{10} + \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}} \right)} < 1$$
    /   1    log(tan(1))\    
    | - -- + -----------|    
    |   10      log(2)  | < 1
atan\2                  /

the solution of our inequality is:
$$x < \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}$$
 _____          
      \    
-------ο-------
       x1
Solving inequality on a graph
Rapid solution 2 [src] 
      log(tan(1)) 
(-oo, -----------)
         log(2)
$$x\ in\ \left(-\infty, \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}\right)$$ 
x in Interval.open(-oo, log(tan(1))/log(2))
Rapid solution [src] 
    log(tan(1))
x < -----------
       log(2)
$$x < \frac{\log{\left(\tan{\left(1 \right)} \right)}}{\log{\left(2 \right)}}$$ 
x < log(tan(1))/log(2)
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