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Math Help - Finding sets of values for given inequalities

  1. #1
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    Question Finding sets of values for given inequalities

    Hi, this is my first thread on this forum, so I don't know if it's in the right place.

    I've recently started sixth form (A-Level) and we are doing Statistics and Calculus. My question is on Calculus.

    In our last lesson we were doing quadratic inequalities, and we got so far as, for example, finding the values of "p" in an equation has real roots.
    The equation would be something like x2+(p+3)x+4p=0, to which the solution is p<=1, p>=9

    This is simple enough for me, but for homework the equations we were given are not so simple. They involve things like multiple inequalities and discriminants which do not factorise.
    Examples of questions I am struggling on are below. Any help would be much appreciated.

    1. What is the set of values for p for which p(x2+2)<2x2+6x+1 for all real values of x?

    2. Find the set of values for k for which x2+3kx+k is positive for all real values of x.

    3. Find the set of values of x for which x2+x+1<x+2<x2-6x+12


    Thanks!
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  2. #2
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    Re: Finding sets of values for given inequalities

    1. What is the set of values for p for which p(x2+2)<2x2+6x+1 for all real values of x?
    px^2 + 2p < 2x^2+6x+1

    (p-2)x^2 -6x + (2p-1) < 0

    note that for the above inequality to be true for all real x , both p-2 < 0 and 36 - 4(p-2)(2p-1) < 0


    2. Find the set of values for k for which x2+3kx+k is positive for all real values of x.
    9k^2 - 4(1)(k) < 0


    3. Find the set of values of x for which x2+x+1<x+2<x2-6x+12
    both inequalities must hold ...

    x^2+x+1 < x+2

    x^2-1 < 0

    and

    x+2 < x^2-6x+12

    0 < x^2-7x+10

    take the intersection of the solution sets
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