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Inequality question

  1. Sep 26, 2005 #1
    A pretty simple problem, but I'm confused nonetheless. :yuck:

    |x-8|<|2x+1|

    Help would be greatly appreciated.
     
  2. jcsd
  3. Sep 26, 2005 #2
    In addition, write |-5x^5y^2| two ways.
     
  4. Sep 26, 2005 #3

    Tide

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    You can start by sketching a graph of each side and comparing them. That should give you some major insights!
     
  5. Sep 26, 2005 #4
    I've pretty much spent hours working on this problem; for some reason, it's still not getting to me. False contradiction and confusion keep popping up...argh
     
  6. Sep 26, 2005 #5
    please help me
     
  7. Sep 26, 2005 #6

    Tide

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    Did you make a sketch?
     
  8. Sep 26, 2005 #7
    ..? Didn't I just explain that I tried this problem for a terriby long time, and I would like a clear explanation?
     
  9. Sep 27, 2005 #8
    Don't we deserve to see your approach?

    |x-8| - |2x+1| < 0

    This solution is based on the fact that if a continuous function changes its sign, it might happen only in this function zeroes.

    1.
    Mark zeroes for each |expression| on the number line: x=8 and x=-1/2.
    This two points break the whole number line into three intervals:
    I. {-inf, -1/2}: -inf < x < 8
    II. [-1/2, 8}: -1/2 <= x < 8
    III. [8, +inf}: 8<= x < +inf.

    Each expression "keeps" its sign unchanged on each interval.
    Draw the number line and put the expressions' signs on the intervals
    x-8: .............-..........|........-.........|.......+.........
    2x+1: ...........-..........|........+.........|.......+.........

    Solve inequality on each region separately following the definition
    |a| = -a, if a<0
    |a| = a, if a>=0

    I.
    x-8<0 and 2x+1<0 therefore
    -(x-8) - (-(2x+1)) < 0
    -x + 8 + 2x + 1 < 0
    x < -9 - agrees with -inf < x < 8 condition.

    II.
    x-8<0 and 2x+1>=0 therefore
    -(x-8) - (2x+1) < 0
    -x + 8 - 2x - 1 < 0
    3x > 7
    x > 7/3 - considering -1/2 <= x < 8 restriction
    the answer on this interval
    7/3 < x < 8

    III.
    x-8>=0 and 2x+1>=0 therefore
    x - 8 - (2x+1) < 0
    x - 8 - 2x - 1 < 0
    x > - 9 - considering x>=8 restriction
    the answer on this interval
    x >= 8.

    Combininig all the answers,
    x < -9 or x > 7/3.

    Faster way - just like Tide recommended.
    Draw two graphs:
    f(x) = |x-8| and g(x) = |2x+1|
    (do you need help with plotting them?).

    |x-8|<|2x+1| is equvalent to
    graph f(x) is below graph g(x).
    These two graphs intersect at x=-9 and x=7/3.
    Graph f(x) is below graph g(x) left of -9 and right of 7/3.
    The same answer.
     
  10. Sep 27, 2005 #9

    HallsofIvy

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    Tide just did!

    Yes, you did. You did not, however, say whether or not one of the things you tried was graphing the two functions as Tide suggested- that was his question.
    In fact, although you say you tried, you have shown us NOTHING of what you tried.
    That is, after all, a requirement for homework help on this forum.
     
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