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Math Help - Simple proving the inequalities

  1. #1
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    Simple proving the inequalities

    |x| ≤ 1 ⇒ |x^2 − x − 2| ≤ 3|x + 1|
    im guessing you factor it and cancel out the x+1 then im left with this
    |x-2| ≤ 3
    What do i do from there?
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  2. #2
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    |x - 2| = |x + (-2)|.

    Now

    |x + (-2)| \leq |x| + |(-2)| by the triangle inequality.

    |x + (-2)| \leq |x| + 2.


    And since |x| \leq 1

    that means |x| + 2 \leq 3.


    Therefore |x - 2| \leq 3

    and thus

    |x| \leq 1 \implies |x^2 - x - 2| \leq 3|x + 1|.
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