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Math Help - Finding values of an inequality to make the expression true

  1. #1
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    Finding values of an inequality to make the expression true

    This is the question:

    If |x -2| < 1, then |2x - 4| < A

    I need to find A. I noticed that the absolute value part has been doubled in the second expression. Naturally, I thought that A had to be double 1, which is 2.

    Thus, A = 2.

    However, my text tells me that the answer is A >= 2. Why is this so? Thank you.
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  2. #2
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    The question asks for what values of A satisfy the equation, right? So yes, you've done all the hard work here, A=2 is a solution that satisfies |2x-4|<A.
    But A=3 is a solution too, is it not? If something is less than 2, then it must certainly be less than 3... and 4... and 2.000000001, and indeed every number greater than 2.
    Thus A>=2.
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  3. #3
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    Hmm, so we know that |2x - 4| < 2. And by your reasoning, |2x - 4| < 3 is also valid. But doesn't that change the expression? Perhaps I'm missing the underlying logic.
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  4. #4
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    Suppose you were asked to choose an integer greater than 13. You might immediately think - "Right, 14 is bigger than 13!". But any number bigger than 14 would have satisfied the criterion too. Any number that you chose that was greater than or equal to 13 would be a valid solution.

    Where do you mean the "expression would change"?
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  5. #5
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    Yes, it "changes the expression". So what? It is still a true expression and that is the whole point!
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  6. #6
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    Suppose you were asked to choose an integer greater than 13. You might immediately think - "Right, 14 is bigger than 13!". But any number bigger than 14 would have satisfied the criterion too. Any number that you chose that was greater than or equal to 13 would be a valid solution.
    Be careful! Nothing was said about "integers". It is certainly true that 13 and 14 are integers but it we restrict this to integers, it in no longer true that "any number bigger than 14 would have satisfied the criterion too".
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  7. #7
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    I wrote this out:

    x < A/2 + 2
    x > -A/2 + 2

    So -A/2 + 2 < x < A/2 + 2

    If A >= 2, and we plotted this on a number line, wouldn't increasing values of A change the range of numbers that x can be?

    (This is why I'm confused)
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  8. #8
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    |x-1|<1

    Hence:

    2|x-1|<2

    |2x-2|<2<A

    A>2
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  9. #9
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    Sorry, I still don't understand.
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