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Solving inequalities/ |z+i|+|z-i|<4

|z+i|+|z-i|<4 inequation

A inequation with variable

The solution

You have entered [src] 
|z + I| + |z - I| < 4
∣z−i∣+∣z+i∣<4\left|{z - i}\right| + \left|{z + i}\right| < 4 
|z - i| + |z + i| < 4
Detail solution
Given the inequality:
∣z−i∣+∣z+i∣<4\left|{z - i}\right| + \left|{z + i}\right| < 4
To solve this inequality, we must first solve the corresponding equation:
∣z−i∣+∣z+i∣=4\left|{z - i}\right| + \left|{z + i}\right| = 4
Solve:
x1=−1.73205080756888x_{1} = -1.73205080756888
x2=1.73205080756888x_{2} = 1.73205080756888
x1=−1.73205080756888x_{1} = -1.73205080756888
x2=1.73205080756888x_{2} = 1.73205080756888
This roots
x1=−1.73205080756888x_{1} = -1.73205080756888
x2=1.73205080756888x_{2} = 1.73205080756888
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}
=
−1.73205080756888+−110-1.73205080756888 + - \frac{1}{10}
=
−1.83205080756888-1.83205080756888
substitute to the expression
∣z−i∣+∣z+i∣<4\left|{z - i}\right| + \left|{z + i}\right| < 4
∣z−i∣+∣z+i∣<4\left|{z - i}\right| + \left|{z + i}\right| < 4
|I + z| + |z - I| < 4

one of the solutions of our inequality is:
x<−1.73205080756888x < -1.73205080756888
 _____           _____          
      \         /
-------ο-------ο-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
x<−1.73205080756888x < -1.73205080756888
x>1.73205080756888x > 1.73205080756888
Rapid solution [src] 
   /   ___            ___\
And\-\/ 3  < x, x < \/ 3 /
−3<x∧x<3- \sqrt{3} < x \wedge x < \sqrt{3} 
(x < sqrt(3))∧(-sqrt(3) < x)
Rapid solution 2 [src] 
    ___    ___ 
(-\/ 3 , \/ 3 )
x in (−3,3)x\ in\ \left(- \sqrt{3}, \sqrt{3}\right) 
x in Interval.open(-sqrt(3), sqrt(3))
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