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Math Help - Solve the following...I keep getting strange answers

  1. #1
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    Solve the following...I keep getting strange answers

    x + \frac{7-2x}{x-2}<= \frac{5x+7}{x-2}

    Solve and express solution as a graph and in interval or set notation.

    The solutions are x=0 or x=9. (That's given.)

    My work:

    he factors I find are x and x-9. Test the factors in the intervals...if their product is - then that interval works. (-inf,0) doesn't work. (0,9) works. (9,inf) doesn't work.

    The problem is the interval (-inf,0) works in the original equation.

    I'm also confused in general with checking for exceptions, where the function is undefined. Is it done before or after simplifying?
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Solve the following...I keep getting strange answers

    You obtain
    \frac{x(x-9)}{x-2}\leq 0

    At this point, we're going to make a sign table:

    x...| ....... 0 .........2 .........9 ...........
    f(x)|........0...........|..........0..........

    Take a test point for example x=1 \Rightarrow f(1)>0 thus:

    x...| ....... 0 .........2 .........9 ...........
    f(x)|....-...0....+.....|...-.....0.....+......

    Therefore the solution of this inequality is:
    ]-\infty,0]\cup ]2,9]
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