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Simple Inequality

  1. Jun 3, 2008 #1
    1. The problem statement, all variables and given/known data
    Solve : [tex]\frac {2}{x} < 3[/tex]

    The answers are [tex]\frac {2}{3} < x and x<0[/tex]

    3. The attempt at a solution

    [tex]\frac {2}{x} = 3[/tex]
    Multiplying by x and dividing by 3, I obtain [tex]\frac {2}{3} < x[/tex]

    Where did they obtain [tex]x<0[/tex] as an answer? Also, accounting that x can be a negative, [tex]\frac {2}{3} > x[/tex] also seems like a solution.
     
  2. jcsd
  3. Jun 3, 2008 #2

    Dick

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    Think about it. If x>0, then you get, as you say, x>2/3 AND x>0. Here the x>0 is superfluous since if x>2/3 it's automatically bigger than 0. If x<0 then you reverse the inequality when you multiply by x, so 2>3x. Or 2/3>x AND x<0. Here the 2/3>x is superfluous, since if x<0 it's automatically less than 2/3. Try some numbers if you don't believe me.
     
  4. Jun 3, 2008 #3
    It just means x either has to be bigger than 2/3 or smaller than 0. Makes sense, for example 1/2 is smaller than 2/3, plug it in and u get 4, which is not smaller than 3. Now try 2 which is bigger than 2/3, u get 1 which is smaller than three. This is why 2/3 > x doesn't make sense.

    Now obviously, if the number on the left was negative it would be smaller than 3 however infinitely small (-1000000) or -0.0001 it is. So anything smaller than 0 would make solve the inequality.
     
  5. Jun 3, 2008 #4

    symbolipoint

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    Dick, you want also to emphasise that x must not be 0; 0 can not be solution in the example.
     
  6. Jun 3, 2008 #5

    Dick

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    I hereby emphasize I do not endorse x=0 as a solution. In fact, I denounce anyone who supports me who would say x=0. Because they would be a terrorist, since 2/0 is not defined. How's that?
     
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