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A natural log inequality with absolute value

  1. Oct 1, 2008 #1
    1. The problem statement, all variables and given/known data

    F(x) = (8-12ln|x|)/(x^4) > 0

    (a) For what values of x is the expression F(x) defined?
    Write your answer in interval notation.

    (b) At what value(s) of x is the expression F(x) equal to zero?
    If there is more than one answer separate them by commas.

    (c) The set of all real numbers x for which the expression
    F(x) is defined and non-zero can be written as the union of several
    mutually disjoint open intervals. Find this set and express
    it as such union.

    (d) By analyzing the sign of F(x) on the above open intervals,
    solve the inequality expressing your answer in interval notation.

    2. The attempt at a solution

    now for (a), i guess since we have |x|, so the domain would be (-inf,0)U(0,inf)
    for (b), i got ln|x| = 2/3 or |x| = e^(2/3) how to get rid of ||?
    (c) .. no idea. and (d) is based on (c).

    Please HELP!!
     
  2. jcsd
  3. Oct 2, 2008 #2

    statdad

    User Avatar
    Homework Helper

    About the point

    [tex]
    \ln |x| = \frac 2 3
    [/tex]

    The positive number whose logarithm is [tex] 2/3 [/tex] is

    [tex]
    e^{2/3}
    [/tex]

    As a hint to answering your question: this fact means that the [tex] x [/tex] values you seek solve

    [tex]
    |x| = e^{2/3}
    [/tex]


    To answer 'c', use the two numbers found above, together with [tex] x = 0 [/tex], to find the three intervals mentioned in part 'c'. Once you know the sign of [tex] F [/tex] at a single spot on each interval, it has that sign throughout, so you can answer the main question.
     
  4. Oct 2, 2008 #3
    first of all thanks for ure help. i didnt get any examples involving ln|x| inequalities anywhere. if u can tell me whre to see such examples, tht wud be nice
    anywys.. here's wht i did:

    b)
    so, if |x| = e^(2/3)
    then x = e^(2/3), -e^(2/3)
    these will be values which make F(x) = 0. right?

    c)so my intervals for which F(x) is "defined and non-zero" wud be (-inf, -e^(2/3)) U (-e^(2/3), 0) U (0, e^(2/3)) U (e^(2/3), inf)

    d) signs wud be - + + -
    and i will choose + signs because F(x) > 0. so answer is (-e^(2/3), 0) U (0, e^(2/3))

    it wud be nice if u can confirm it! :)
    THANKS AGAIN!!
     
  5. Oct 2, 2008 #4

    statdad

    User Avatar
    Homework Helper

    Looks good.
     
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