Algebra II- please help!

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I just cannot seem to be able to grasp the concept of solving polynomial inequalities. Can someone please explain, step by step, how to solve them? Here's a problem I can't solve. Please use this as an example:

(x-2)(x-5)<0

I cannot thank you enough for helping me with this. I have a test on this tomorrow and this is the only section I don't get. Thank you for all the help!

  • Algebra II- please help! -

    First of all, sketch a graph of
    y = (x-2)(x-5)
    This will be a parabola because it is a quadratic, a x^2 + b x + c
    for large negative x, y is big positive
    it is zero at x = 2
    then it is negative for a while
    it is then zero at x = 5
    then it goes big positive
    so
    negative between x = 2 and x = 5

  • Algebra II- please help! -

    Thank you so much, Damon! You really helped me out! : )

  • Algebra II- please help! -

    I used to teach it with a slightly different approach because it worked well with multiple factors.
    for example I will add another factor to your problem

    (x-2)(x-5)(x+2) > 0

    the x-intercepts of the corresponding function would be x = -2,2, and 5

    mark those on the x-axis

    there are now 4 sections to my x-axis
    1. all values below -2
    2. all values between -2 and 2
    3. all values between 2 and 5
    4. all values greater than 5

    now pick an arbitrary value in each section and test it. You don't have to actually evaluate it, just worry about + or - (matching the > or <)

    1. let x=-5 ---- (-)(-)(-) = - (does not work)
    2. let x = 0 ---(-)(-)(+) = + works
    3. let x = 3 ---(+)(-)(+) = - no
    4. let x = 6 --- (+)(+)(+) = + yes

    so -2 < x < 2 OR x > 5

    for quadratic inequalities, I would also use the method outlined by Damon, but for cubics and higher, the above method works very nicely.
    Of course if the function does not factor it would be a mess anyway.

  • Algebra II- please help! -

    Thank you, Reiny. My teacher taught me something very similar to your method and now I understand it. Thank you for your help! : )

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