Mister Exam

Other calculators

Solving inequalities/ tgx<2

tgx<2 inequation

A inequation with variable

The solution

You have entered [src] 
tan(x) < 2
tan⁡(x)<2\tan{\left(x \right)} < 2 
tan(x) < 2
Detail solution
Given the inequality:
tan⁡(x)<2\tan{\left(x \right)} < 2
To solve this inequality, we must first solve the corresponding equation:
tan⁡(x)=2\tan{\left(x \right)} = 2
Solve:
Given the equation
tan⁡(x)=2\tan{\left(x \right)} = 2
- this is the simplest trigonometric equation
This equation is transformed to
x=πn+atan⁡(2)x = \pi n + \operatorname{atan}{\left(2 \right)}
Or
x=πn+atan⁡(2)x = \pi n + \operatorname{atan}{\left(2 \right)}
, where n - is a integer
x1=πn+atan⁡(2)x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}
x1=πn+atan⁡(2)x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}
This roots
x1=πn+atan⁡(2)x_{1} = \pi n + \operatorname{atan}{\left(2 \right)}
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x1x_{0} < x_{1}
For example, let's take the point
x0=x1−110x_{0} = x_{1} - \frac{1}{10}
=
(πn+atan⁡(2))+−110\left(\pi n + \operatorname{atan}{\left(2 \right)}\right) + - \frac{1}{10}
=
πn−110+atan⁡(2)\pi n - \frac{1}{10} + \operatorname{atan}{\left(2 \right)}
substitute to the expression
tan⁡(x)<2\tan{\left(x \right)} < 2
tan⁡(πn−110+atan⁡(2))<2\tan{\left(\pi n - \frac{1}{10} + \operatorname{atan}{\left(2 \right)} \right)} < 2
tan(-1/10 + pi*n + atan(2)) < 2

the solution of our inequality is:
x<πn+atan⁡(2)x < \pi n + \operatorname{atan}{\left(2 \right)}
 _____          
      \    
-------ο-------
       x1
Solving inequality on a graph
0-80-60-40-2020406080-2500025000
Rapid solution [src] 
  /                             /         pi    \\
Or|And(0 <= x, x < atan(2)), And|x <= pi, -- < x||
  \                             \         2     //
(0≤x∧x<atan⁡(2))∨(x≤π∧π2<x)\left(0 \leq x \wedge x < \operatorname{atan}{\left(2 \right)}\right) \vee \left(x \leq \pi \wedge \frac{\pi}{2} < x\right) 
((0 <= x)∧(x < atan(2)))∨((x <= pi)∧(pi/2 < x))
Rapid solution 2 [src] 
                pi     
[0, atan(2)) U (--, pi]
                2
x in [0,atan⁡(2))∪(π2,π]x\ in\ \left[0, \operatorname{atan}{\left(2 \right)}\right) \cup \left(\frac{\pi}{2}, \pi\right] 
x in Union(Interval.Ropen(0, atan(2)), Interval.Lopen(pi/2, pi))
    © Mister Exam – Calculator