Question

Find the set of points formed by the expression π<|π§β4+2π|<3π.

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Hermann

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To find the set of points formed by the expression π<|π§β4+2π|<3π, we need to solve the inequality.

Let π§ = π₯ + π¦π, where π₯ and π¦ are real numbers.

The inequality can be written as:

π < |(π₯ + π¦π) β (4 + 2π)| < 3π

Simplifying the expression inside the absolute value:

π < |π₯ β 4 + (π¦ β 2)π| < 3π

Let π = π₯ β 4 and π = π¦ β 2. The inequality becomes:

π < |π + ππ| < 3π

Using the polar representation of complex numbers, π + ππ can be written as:

π + ππ = π(πππ (π) + ππ ππ(π))

where π = |π + ππ| is the magnitude of π + ππ and π is the argument of π + ππ.

We can rewrite the inequality as:

π < |π(πππ (π) + ππ ππ(π))| < 3π

Since π is always nonnegative, we can remove the absolute value and rewrite the inequality as:

π < π < 3π

Therefore, the set of points formed by the expression π < |π§β4+2π| < 3π is the set of all complex numbers π§ such that the magnitude of π§ β (4 + 2π) is between π and 3π.

Answer: The set of points formed by the expression π < |π§β4+2π| < 3π is π < π < 3π, where π§ = π₯ + π¦π and π is the magnitude of π§ β (4 + 2π).

Let π§ = π₯ + π¦π, where π₯ and π¦ are real numbers.

The inequality can be written as:

π < |(π₯ + π¦π) β (4 + 2π)| < 3π

Simplifying the expression inside the absolute value:

π < |π₯ β 4 + (π¦ β 2)π| < 3π

Let π = π₯ β 4 and π = π¦ β 2. The inequality becomes:

π < |π + ππ| < 3π

Using the polar representation of complex numbers, π + ππ can be written as:

π + ππ = π(πππ (π) + ππ ππ(π))

where π = |π + ππ| is the magnitude of π + ππ and π is the argument of π + ππ.

We can rewrite the inequality as:

π < |π(πππ (π) + ππ ππ(π))| < 3π

Since π is always nonnegative, we can remove the absolute value and rewrite the inequality as:

π < π < 3π

Therefore, the set of points formed by the expression π < |π§β4+2π| < 3π is the set of all complex numbers π§ such that the magnitude of π§ β (4 + 2π) is between π and 3π.

Answer: The set of points formed by the expression π < |π§β4+2π| < 3π is π < π < 3π, where π§ = π₯ + π¦π and π is the magnitude of π§ β (4 + 2π).

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