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Find the set of points formed by the expression πœ‹<|π‘§βˆ’4+2𝑖|<3πœ‹.

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Answer to a math question Find the set of points formed by the expression πœ‹<|π‘§βˆ’4+2𝑖|<3πœ‹.

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Hermann
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To find the set of points formed by the expression πœ‹<|π‘§βˆ’4+2𝑖|<3πœ‹, we need to solve the inequality.

Let 𝑧 = π‘₯ + 𝑦𝑖, where π‘₯ and 𝑦 are real numbers.

The inequality can be written as:
πœ‹ < |(π‘₯ + 𝑦𝑖) βˆ’ (4 + 2𝑖)| < 3πœ‹

Simplifying the expression inside the absolute value:
πœ‹ < |π‘₯ βˆ’ 4 + (𝑦 βˆ’ 2)𝑖| < 3πœ‹

Let π‘Ž = π‘₯ βˆ’ 4 and 𝑏 = 𝑦 βˆ’ 2. The inequality becomes:
πœ‹ < |π‘Ž + 𝑏𝑖| < 3πœ‹

Using the polar representation of complex numbers, π‘Ž + 𝑏𝑖 can be written as:
π‘Ž + 𝑏𝑖 = π‘Ÿ(π‘π‘œπ‘ (πœƒ) + 𝑖𝑠𝑖𝑛(πœƒ))

where π‘Ÿ = |π‘Ž + 𝑏𝑖| is the magnitude of π‘Ž + 𝑏𝑖 and πœƒ is the argument of π‘Ž + 𝑏𝑖.

We can rewrite the inequality as:
πœ‹ < |π‘Ÿ(π‘π‘œπ‘ (πœƒ) + 𝑖𝑠𝑖𝑛(πœƒ))| < 3πœ‹

Since π‘Ÿ is always nonnegative, we can remove the absolute value and rewrite the inequality as:
πœ‹ < π‘Ÿ < 3πœ‹

Therefore, the set of points formed by the expression πœ‹ < |π‘§βˆ’4+2𝑖| < 3πœ‹ is the set of all complex numbers 𝑧 such that the magnitude of 𝑧 βˆ’ (4 + 2𝑖) is between πœ‹ and 3πœ‹.

Answer: The set of points formed by the expression πœ‹ < |π‘§βˆ’4+2𝑖| < 3πœ‹ is πœ‹ < π‘Ÿ < 3πœ‹, where 𝑧 = π‘₯ + 𝑦𝑖 and π‘Ÿ is the magnitude of 𝑧 βˆ’ (4 + 2𝑖).

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