Mister Exam
|z+i|+|z-i|<4 inequation
The solution
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|z + I| + |z - I| < 4
$$\left|{z - i}\right| + \left|{z + i}\right| < 4$$
|z - i| + |z + i| < 4
Detail solution
Given the inequality:
$$\left|{z - i}\right| + \left|{z + i}\right| < 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{z - i}\right| + \left|{z + i}\right| = 4$$
Solve:
$$x_{1} = -1.73205080756888$$
$$x_{2} = 1.73205080756888$$
$$x_{1} = -1.73205080756888$$
$$x_{2} = 1.73205080756888$$
This roots
$$x_{1} = -1.73205080756888$$
$$x_{2} = 1.73205080756888$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1.73205080756888 + - \frac{1}{10}$$
=
$$-1.83205080756888$$
substitute to the expression
$$\left|{z - i}\right| + \left|{z + i}\right| < 4$$
$$\left|{z - i}\right| + \left|{z + i}\right| < 4$$
one of the solutions of our inequality is:
$$x < -1.73205080756888$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -1.73205080756888$$
$$x > 1.73205080756888$$
$$\left|{z - i}\right| + \left|{z + i}\right| < 4$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{z - i}\right| + \left|{z + i}\right| = 4$$
Solve:
$$x_{1} = -1.73205080756888$$
$$x_{2} = 1.73205080756888$$
$$x_{1} = -1.73205080756888$$
$$x_{2} = 1.73205080756888$$
This roots
$$x_{1} = -1.73205080756888$$
$$x_{2} = 1.73205080756888$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1.73205080756888 + - \frac{1}{10}$$
=
$$-1.83205080756888$$
substitute to the expression
$$\left|{z - i}\right| + \left|{z + i}\right| < 4$$
$$\left|{z - i}\right| + \left|{z + i}\right| < 4$$
|I + z| + |z - I| < 4
one of the solutions of our inequality is:
$$x < -1.73205080756888$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -1.73205080756888$$
$$x > 1.73205080756888$$
Rapid solution
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/ ___ ___\ And\-\/ 3 < x, x < \/ 3 /
$$- \sqrt{3} < x \wedge x < \sqrt{3}$$
(x < sqrt(3))∧(-sqrt(3) < x)
Rapid solution 2
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___ ___ (-\/ 3 , \/ 3 )
$$x\ in\ \left(- \sqrt{3}, \sqrt{3}\right)$$
x in Interval.open(-sqrt(3), sqrt(3))