Mister Exam
|z-2+i|>3 inequation
The solution
Detail solution
Given the inequality:
$$\left|{z - 2 + i}\right| > 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{z - 2 + i}\right| = 3$$
Solve:
$$x_{1} = 4.82842712474619$$
$$x_{2} = -0.82842712474619$$
$$x_{1} = 4.82842712474619$$
$$x_{2} = -0.82842712474619$$
This roots
$$x_{2} = -0.82842712474619$$
$$x_{1} = 4.82842712474619$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-0.82842712474619 - \frac{1}{10}$$
=
$$-0.92842712474619$$
substitute to the expression
$$\left|{z - 2 + i}\right| > 3$$
$$\left|{z - 2 + i}\right| > 3$$
one of the solutions of our inequality is:
$$x < -0.82842712474619$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -0.82842712474619$$
$$x > 4.82842712474619$$
$$\left|{z - 2 + i}\right| > 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{z - 2 + i}\right| = 3$$
Solve:
$$x_{1} = 4.82842712474619$$
$$x_{2} = -0.82842712474619$$
$$x_{1} = 4.82842712474619$$
$$x_{2} = -0.82842712474619$$
This roots
$$x_{2} = -0.82842712474619$$
$$x_{1} = 4.82842712474619$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-0.82842712474619 - \frac{1}{10}$$
=
$$-0.92842712474619$$
substitute to the expression
$$\left|{z - 2 + i}\right| > 3$$
$$\left|{z - 2 + i}\right| > 3$$
|-2 + I + z| > 3
one of the solutions of our inequality is:
$$x < -0.82842712474619$$
_____ _____
\ /
-------ο-------ο-------
x_2 x_1Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -0.82842712474619$$
$$x > 4.82842712474619$$
Rapid solution
[src]
/ / ___\ / ___ \\ Or\And\-oo < x, x < 2 - 2*\/ 2 /, And\x < oo, 2 + 2*\/ 2 < x//
$$\left(-\infty < x \wedge x < - 2 \sqrt{2} + 2\right) \vee \left(x < \infty \wedge 2 + 2 \sqrt{2} < x\right)$$
((-oo < x)∧(x < 2 - 2*sqrt(2)))∨((x < oo)∧(2 + 2*sqrt(2) < x))
Rapid solution 2
[src]
___ ___ (-oo, 2 - 2*\/ 2 ) U (2 + 2*\/ 2 , oo)
$$x\ in\ \left(-\infty, - 2 \sqrt{2} + 2\right) \cup \left(2 + 2 \sqrt{2}, \infty\right)$$
x in Union(Interval.open(-oo, 2 - 2*sqrt(2)), Interval.open(2 + 2*sqrt(2), oo))