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           ☆1 inequalities

[ 1/1Inequality]
    Assignment:Find the solution set of inequality 0 <(-x^2+2x+3)/(sqrt(x^2+1)*sqrt(5x^2+12x+9)) <1 .
    Question type: Inequality
    Solution:
    The inequality can be reduced to 2 inequalities：
        0 < ( -x ^ 2 + 2 * x + 3 ) / ( sqrt ( x ^ 2 + 1 ) * sqrt ( 5 * x ^ 2 + 12 * x + 9 ) )         (1)
         ( -x ^ 2 + 2 * x + 3 ) / ( sqrt ( x ^ 2 + 1 ) * sqrt ( 5 * x ^ 2 + 12 * x + 9 ) ) <1         (2)
        From the definition field of √
         x ^ 2 + 1 ≥ 0        (3 )
        From the definition field of √
         5 * x ^ 2 + 12 * x + 9 ≥ 0        (4 )
        From the definition field of divisor
         sqrt ( x ^ 2 + 1 ) * sqrt ( 5 * x ^ 2 + 12 * x + 9 ) ≠ 0        (5 )

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    From inequality(1)：
         -1 < x < 3
    From inequality(2)：
         x < 0 或  x ＞ 0
    From inequality(3)：
         x ∈ R (R is all real numbers),that is, in the real number range, the inequality is always established!
    From inequality(4)：
         x ∈ R (R is all real numbers),that is, in the real number range, the inequality is always established!
    From inequality(5)：
         x ∈ R （R为全体实数），即在实数范围内，不等式恒成立！

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    From inequalities (1) and (2)
         -1 < x < 0 或  0 < x < 3     (6)
    From inequalities (3) and (6)
         -1 < x < 0 或  0 < x < 3     (7)
    From inequalities (4) and (7)
         -1 < x < 0 或  0 < x < 3     (8)
    From inequalities (5) and (8)
         -1 < x < 0 或  0 < x < 3     (9)

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    The final solution set is :
         -1 < x < 0 或  0 < x < 3

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