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Math Help - Solving inequality

  1. #1
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    Solving inequality

    See this example:

    \frac{x - 1}{x + 1} \geq \frac{2x}{x - 1}

    We multiply both sides by  (x + 1) (x - 1)

    and get:

    (x - 1)^2 \geq 2x(x +1)

    Criteria 1:

    x < -1
    (x - 1)^2 \geq 2x(x +1)
    x^2 - 2x + 1 \geq 2x^2 + 2x
    x^2 + 4x -1 \leq 0

    x_{1,2} = \frac{-4 \pm 2\sqrt{5}}{2} = -2 \pm \sqrt{5}

    x \in [-2-\sqrt{5}, -2+\sqrt{5}] but since  x < -1 we get x \in [-2-\sqrt{5}, 1)

    Criteria 2:
     x \in (-1, 1)

    (x - 1)^2\leq 2x(x +1) why less or equal??

    from that we get:  x^2 +4x -1 \geq 0 \Rightarrow x \in [-2+\sqrt{5},1) (why not (-1, -2+\sqrt{5}] ?)

    Criteria 3:
    x > 1
     x^2 + 4x - 1 \leq 0 \Rightarrow no solution

    All together:

     x \in [-2-\sqrt{5},-1) U [-2+\sqrt{5},1)
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  2. #2
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    If x \in (-1,1), then (x - 1)(x + 1) is negative.
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  3. #3
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    short and simple. thanks
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