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x^3-2x^2-9x-2 is greater than or equal to -20. I got (-9+x^2)(x-2)(x-2). That would mean for the first grouping, the answer would be -3 or 3. My book listed them separately. Could I instead go +/- 3? I then got double root of two. But my book says the other answer is infinity. Where's that come from? Thanks!!!!!!!!!

  • Algebra -

    The given inequality is
    x^3-2x^2-9x-2 ≥ -20
    f(x)=x^3-2x^2-9x-18 ≥ 0

    F(x) can be factored into (x-3)*(x-2)*(x+3) so the roots of f(x) are +3, -3 and 2.

    F(x) has a positive leading coefficient, so the graph goes to -∞ on the left, and to +∞ on the right.

    We can therefore deduce that for x<-3, f(x) is negative, and for x>3, f(x) is positive.
    Also, since the intermediate root is x=2, we can also deduce that f(x) is non-negative between x=-3 and 2.
    So the solution set for the given inequality is:
    [-3,2]∪[3,∞), or alternatively,
    -3≤x≤2 or x≥3.

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