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For which real values of x does the inequality log base 0.2 of ((x + 2)/x) ≤ 1 hold true?

1. (-∞, -5/2] ∪ (0, ∞)
2. [5/2, ∞)
3. (-∞, -2) ∪ [0, ∞)
4. none of these

Correct answer: (-∞, -5/2] ∪ (0, ∞)

Solution

The inequality log base 0.2 of ((x + 2)/x) ≤ 1 holds true for values of x where the expression inside the logarithm is positive and the logarithm itself is less than or equal to 1. This occurs for x in the intervals (-∞, -5/2] and (0, ∞), as these intervals ensure the argument of the logarithm is valid and satisfies the inequality.

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