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Rewriting inequality help

  1. Mar 19, 2007 #1


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    Solve the inequality [tex]
    \left| {x - 9} \right| - \left| x \right| \ge 9

    I started by rewriting it as:
    \left| {x - 9} \right| \ge 9 + \left| x \right|

    Now, for any real numbers x and y,
    \left| {x + y} \right| \le \left| x \right| + \left| y \right|

    According to that,
    |x-9| cannot be greater than |x|+9, but it can be equal, if x and and 9 are of the same sign. The 9 is negative, so the x must also be negative, giving the solution [tex]
    x \le 0

    Why is it that the answer book says the answer is [tex]
    x < 0[/tex]? Why is it excluding zero? Or is it just wrong?

  2. jcsd
  3. Mar 19, 2007 #2


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    All I can suggest is that you go back and check the problem again. If there really is a "[itex]\ge[/itex]" sign rather than just >, obviously x= 0 is a solution. Either you miscopied the problem or the book's answer is wrong.
  4. Mar 19, 2007 #3


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    I have definitly copied the problem correctly from the book.
  5. Mar 19, 2007 #4
    This would have many solutions... first x < 0, 0< x < 9 and with x > 9.
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