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решите неравенство
[tex] \sqrt{5x { }^{2} + 6x + 1} \geqslant \frac{x}{ \sqrt{x {}^{2} } } [/tex]
решите неравенство
[tex] \sqrt{5x { }^{2} + 6x + 1} \geqslant \frac{x}{ \sqrt{x {}^{2} } } [/tex]
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Verified answer
√(5x² + 6x + 1) ≥ x/√x²
5x² + 6x + 1 = 0
D = 36 - 20 = 16
x12 = (-6 +- 4)/10 = -1 -1/5
x ≠ 0
одз x ∈ (-∞ -1] U [-1/5 0) U (0 +∞)
√(5x + 1)(x + 1) ≥ x/|x|
x ≥ 0
√(5x + 1)(x + 1) ≥ 1
5x² + 6x + 1 ≥ 1
x(5x + 6) ≥ 0
x ∈ (-∞ -5/6] U [0, +∞)
пересекаем с одз x ∈ (0 +∞)
x < 0
√(5x + 1)(x + 1) ≥ -1
x ∈ (-∞ -1] U [-1/5 0)
объединяем решения
x ∈ (-∞, -1] U [-1/5, 0) U (0, +∞)