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Find all real x satisfying |1 + 4i - 2^(-x)| <= 5, where i = sqrt(-1).

1. [-2, infinity)
2. (-infinity, 2]
3. [0, infinity)
4. [-2, 0]

Correct answer: [-2, infinity)

Solution

The condition becomes (1 - 2^(-x))² + 16 <= 25, i.e. (1 - 2^(-x))² <= 9, giving -2 <= 2^(-x) <= 4; since 2^(-x) > 0 this means 2^(-x) <= 4, i.e. -x <= 2, so x >= -2.

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