math ergent please please...

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10.a)Solve the inequality x^3-5x^2+2x+8<0 by

i) using intervals
ii) considering all cases

I did this but my teacher said that when i solved this: x^3-5x^2+2x+8<0
(x+1)(x^2-6x+8)
(x+1)(x-4)(x-2)<0

She told me to show all the steps, but i don't know how to show more. so can you please help me to show all the works and rest of the works i did too but she said " the same answer has to happen for each method. i don't really understand that. can you help me please and thank you....

Steve explained this, but still i didn't got the steps. can someone do it for me step by step so i can understand the steps of it please.

  • math ergent please please... -

    Ok, so all the algebra work has been done
    x^3-5x^2+2x+8<0
    (x+1)(x-4)(x-2)<0

    Quickest way:
    I would look at the graph of
    y = x^3-5x^2+2x+8
    = (x+1)(x-4)(x-2)
    I know that this is standard cubic function opening up in quadrant I and having x-intercepts of -1, 2, and 4
    So a very rough sketch shows me that this curve is below the x-axis ( < 0 ) for x < - 1 or 2 < x < 4

    looking at all cases , the critical values of -2,2, and 4 split the line into 4 parts :

    if x < -1, try some value, e.g. x = -5
    (-)(-)(+) < 0 , so yes
    x < -1

    if x is between -2 and 2 , e.g. x = 0
    (+)(-)(-2) > 0 , so NO

    if x is between 2 and 4 , e.g. x = 3
    (+)(-)(+) < 0 , so YES
    x is between 2 and 4

    if x > 4 , e.g. x = 5
    (+)(+)(+) > 0 , NO

    then x < -2 OR 2 < x < 4

  • math ergent please please... -

    Reiny help me with using intervals. u forgot to help me with that. its urgent please

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