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Math Help - quadratic inequality

  1. #1
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    quadratic inequality

    x^2-2x-8>0

    I know that the answer is x>4 or x<-2. My question is why the sign is reversed if there is no multiplication or division on either side?
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  2. #2
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    Consider x^2-2x-8 = (x-4)(x+2). This is > 0 if either x-4 and x+2 are both > 0 or both < 0. The first case leads to x>4 and x>-2, which is just x>4; the second to x<4 and x<-2, which is x<-2. The reversal of sign comes from this second case where you consider the possibility that two negative factors multiply to give a positive value.
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  3. #3
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    Here is one way.

    Get the critical points of the inequality, or the points when the graph of the inequality crosses the x-axis. You treat the inequality as an equation only---meaning, forget about the < and > signs. Consider only the = sign.
    Then, these critical points subdivides the x-axis into intervals of the x-values.
    Then, test/check the inequality in these intervals. The solution/solutions of the inequality are the intervals where the inequality is valid/true.

    x^2 -2x -8 > 0

    x^2 -2x -8 = 0
    (x-4)(x+2) = 0
    x = 4, or -2

    So there are 3 inervals:
    (-infinity,-2), (-2,4), and (4,infinity)

    Test the x^2 -2x -8 > 0 in the (-infinity,-2) interval.
    Say, x = -3.
    (-3)^2 -2(-3) -8 >? 0
    9 +6 -8 >? 0
    7 >? 0
    Yes, so, this interval is a solution. -----***

    Test the x^2 -2x -8 > 0 in the (-2,4) interval.
    Say, x = 0.
    (0)^2 -2(0) -8 >? 0
    0 -0 -8 >? 0
    -8 >? 0
    No, so, this interval is not a solution.

    Test the x^2 -2x -8 > 0 in the (4,infinity) interval.
    Say, x = 7.
    (7)^2 -2(7) -8 >? 0
    49 -14 -8 >? 0
    27 >? 0
    Yes, so, this interval is a solution also. -----***

    Therefore,
    x = (-infinity,-2) ---or x < -2
    x = (4, infinity) ----or x > 4

    ---------------
    What about the critical x's themselves, when x = -2, and when x=4?

    The inequality does not say "greater than or equal to" zero, hence x cannot equal -2 0r 4.

    If the inequality were x^2 -2x -8 >= 0,
    then x <= -2 or x >= 4.
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