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Math Help - inequality involving a fraction

  1. #1
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    inequality involving a fraction

    x-1/x+2> or = 0
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  2. #2
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    Firslty you want to identify the points at which the expression is equal to zero. That is, solve x-1/x+2 = 0. Multiplying up by x gives x^2 + 2x - 1 = 0, which has solutions x = -1 + sqrt(2) and -1 - sqrt(2), approximately 0.414 and -2.414 respectively. This particular expression also has a vertical asymptote, a polite way of saying there's an infinity, at x=0. So the expression isn't going to cross the x-axis, that is, isn't going to change sign, except at these three points. So let's see what its values are like in the regions

    x < -1-sqrt(2) for example x = -10 value = -7.9 sign is -
    x = -1-sqrt(2) value = 0
    -1-sqrt(2) < x < 0 for example x = -1 value = 4 sign is +
    x = 0 value undefined
    0 < x < -1+sqrt(2) for example x = 0.1 value = -7.9 sign is -
    x = -1+sqrt(2) value = 0
    -1+sqrt(2) < x for example x = 10 value = 11.9 sign is +

    From this table you can read off the ranges of values for which the expression is >= 0.
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