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math

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how do you solve this inequality
3x-2 < x+4/x-2
please help i don't get it

  • math -

    You must mean:
    3x-2 < (x+4)/(x-2) , so clearly x ≠ 2
    (x-2)(3x-2) < x+4
    3x^2 - 8x + 4 - x - 4 < 0
    3x^2 - 9x < 0
    3x(x-3) < 0

    So the "critical values" are x=0 , x = 2, and x=3

    test for a value of x<0
    let x=-5 in original
    -17 < -1/-7
    -17 < 1/7 ? YES

    test for a value between 0 and 2
    let x=1
    1 < 5/-1
    1 < -5 ? NO

    test for a value between 2 and 3
    let x = 2.5
    5.5 < 6.5/.5
    5.5 < 13 ? , YES

    test for a value x>3
    let x=5
    13 < 9/3
    13 < 3 ? , NO

    so we have x ≤ 0 OR 2 < x ≤ 3 , notice that x=2 in not included

    To see that this is correct
    go to my favourite graphing program
    http://rechneronline.de/function-graphs/
    and enter
    3x-2 - (x+4)/(x-2)

    in the window for "first graph"
    change the "Range y-axis from" entries to -200 to 200

    you will see the curve below the x-axis from -infinitity to 0
    above the x-axis from 0 to the asymptote of 2
    below the x-axis from 2 to 3, and
    above the x-axis for x>3

  • math -

    thanks so much :)

  • math -

    what if all the values you put in do not work. how would you write the intervals.?

  • math -

    If the inequality can not be satisfied for any real value of x, then you just have to say that.

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