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Let alpha and beta be the roots of x² + alpha*x + beta = 0, with alpha not equal to beta. Which of the following statements about the inequality ||x - beta| - alpha| < alpha are correct?

1. The inequality is satisfied by exactly two integral values of x
2. The inequality is satisfied by all values of x in (-4, -2)
3. The roots of the equation are opposite in sign
4. x² + alpha*x + beta < 0 for all x in [-1, 0]

Correct answer: The inequality is satisfied by all values of x in (-4, -2)

Solution

From alpha+beta = -alpha: beta = -2*alpha. From alpha*beta = beta: if beta ≠ 0, then alpha = 1 (dividing both sides by beta, and -2*alpha² = -2*alpha gives alpha = 1). So alpha = 1, beta = -2. The inequality becomes ||x+2|-1| < 1, i.e. 0 < |x+2| < 2, i.e. x in (-4,-2) union (-2, 0). This is satisfied by all x in (-4,-2) — statement B is correct. Integral values in (-4,0) excluding -2: x = -3 and x = -1, giving exactly two integral values — statement A is also correct. The roots are 1 and -2 (opposite signs) — C is correct. For x in [-1,0]: f(-1) = 1-1-2=-2 < 0, f(0) = 0-0-2=-2 < 0; since the quadratic opens upward (positive leading coeff), f(x) < 0 between the roots (x in (-2,1)), so all x in [-1,0] satisfy it — D is correct. All four statements are correct.

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