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2\sin x + 3\cos x + 4 \in [4 - \sqrt{13}, 4 + \sqrt{13}]
Since $ \sqrt{13} \approx 3.605 $, $ 4 - \sqrt{13} \approx 0.395 > 0 $, so the expression never reaches zero.
Therefore, the absolute value $ |2\sin x + 3\cos x + 4| $ achieves its minimum at the minimum of the continuous function on an interval where it is always positive.

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