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Math Help - Fraction inequality denominator 0 rule?

  1. #1
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    Fraction inequality denominator 0 rule?

    -x+8
    ----- (greater than or equal) 0
    x-7

    Do the problem I get x=8 and x=7

    I graph it, and test it, and am wondering what exactly is the rule for when to put () or []

    My understanding is that since it is a (greater than or equal) sign it should be []

    But, the answer states it is (7,8] and Im not sure why.

    I read something that if a number in the solution makes the denominator 0 then you must include it with () and I would have gotten it wrong as I was going to enter it as [7,8]

    Is it basically this? if a number that is the solution on my graph makes the original equations denominator 0 then that number will have a () by it?

    Maybe someone can dumb it down for me
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  2. #2
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    Re: Fraction inequality denominator 0 rule?

    The general fact is that \frac{a}{b}\ge0 iff (a ≥ 0 and b > 0) or (a ≤ 0 and b < 0). Yes, the inequality for b is strict because the denominator cannot become zero.

    Applying this fact to this problem, we get (1) -x + 8 ≥ 0 and x - 7 > 0 or (2) -x + 8 ≤ 0 and x - 7 < 0. Variant (1) is equivalent to 7 < x ≤ 8, i.e., x ∈ (7, 8]. Variant (2) is equivalent to 8 ≤ x < 7, which is impossible.
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  3. #3
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    Re: Fraction inequality denominator 0 rule?

    So in dummer terms, for me, I need it simple, is it true then that any time I get an answer with numbers that make the denominator 0, use () with that number when writing the answer?
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  4. #4
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    Re: Fraction inequality denominator 0 rule?

    Quote Originally Posted by itgl72 View Post
    So in dummer terms, for me, I need it simple, is it true then that any time I get an answer with numbers that make the denominator 0, use () with that number when writing the answer?
    Yes, if a number that makes a denominator 0 appears as the end of an interval in the final answer, then it must be excluded by using a parenthesis instead of a square bracket.
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  5. #5
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    Re: Fraction inequality denominator 0 rule?

    (a, b) means the points a and b are NOT included in the set.
    (a, b] means the point a is not included but b is.
    [a, b) means the point a is included but b is not.
    [a, b] means the points a and b are both included in the set.

    Put the endpoints into the original inequality to see if they satify it or not.
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