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Thread: Positive a/b

  1. #1
    Super Member harpazo's Avatar
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    Positive a/b

    If the quotient a/b is positive, which of the following must be true?

    A. a > 0
    B. b > 0
    C. ab > 0
    D. a - b > 0
    E. a + b > 0

    My Work:

    If the given quotient is positive, then a/b > 0.
    So, the following is possible:

    a/b > 0

    b(a/b) > b0

    Conclusion: a > 0 is my answer.
    The book's answer is C. Why?
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    Re: Positive a/b

    Quote Originally Posted by harpazo View Post
    If the quotient a/b is positive, which of the following must be true?

    A. a > 0
    B. b > 0
    C. ab > 0
    D. a - b > 0
    E. a + b > 0

    My Work:

    If the given quotient is positive, then a/b > 0.
    So, the following is possible:

    a/b > 0

    b(a/b) > b0

    Conclusion: a > 0 is my answer.
    The book's answer is C. Why?
    If $a=-4~\&~b=-3$ it is true that $\dfrac{-4}{-3}>0$ now you check out each in that list.
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    Re: Positive a/b

    Please recall that multiplying both sides of an inequality, like $\displaystyle \frac{a}{b}> 1$, by a retains the direction of the inequality only if a is positive. If a is negative, the inequality is reversed.

    Given that $\displaystyle \frac{a}{b}> 0$ and a is positive then $\displaystyle \left(\frac{b}{a}\right)a= b> 0(a)= 0$ so b is also positive. But if a is negative then $\displaystyle \left(\frac{a}{b}\right)a= b< 0(a)= 0$. That is, if $\displaystyle \frac{b}{a}> 0$ then either a and b are both positive or a and b are both negative.
    Last edited by topsquark; Feb 5th 2019 at 07:41 AM. Reason: Tweaked LaTeX
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    Super Member harpazo's Avatar
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    Re: Positive a/b

    Quote Originally Posted by HallsofIvy View Post
    Please recall that multiplying both sides of an inequality, like [tex]\frac{a}{b}> 1[tex], by a retains the direction of the inequality only if a is positive. If a is negative, the inequality is reversed.

    Given that $\displaystyle \frac{a}{b}> 0$ and a is positive then $\displaystyle \left(\frac{b}{a}\right)a= b> 0(a)= 0$ so b is also positive. But if a is negative then [tex]\left(\frac{a}{b}\right)a= b< 0(a)= 0[tex]. That is, if $\displaystyle \frac{b}{a}> 0$ then either a and b are both positive or a and b are both negative.
    Your LaTex work does not display. For the LaTex that is displayed, the words block your reply.
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    Super Member harpazo's Avatar
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    Re: Positive a/b

    Quote Originally Posted by Plato View Post
    If $a=-4~\&~b=-3$ it is true that $\dfrac{-4}{-3}>0$ now you check out each in that list.
    Check out each in the list by letting a = -4 and b = -3?
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    Re: Positive a/b

    Quote Originally Posted by harpazo View Post
    Check out each in the list by letting a = -4 and b = -3?
    For example: E. says a+b>0.
    If a=-4 & b=-3 then -4-3=-7 so E. is FALSE.
    Do each of A B C D.
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    Super Member harpazo's Avatar
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    Re: Positive a/b

    Quote Originally Posted by Plato View Post
    For example: E. says a+b>0.
    If a=-4 & b=-3 then -4-3=-7 so E. is FALSE.
    Do each of A B C D.
    a = -4, b = -3

    ab > 0

    (-4)(-3) > 0

    12 > 0

    True statement. Thus, C is correct.
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    Re: Positive a/b

    Quote Originally Posted by harpazo View Post
    a = -4, b = -3
    ab > 0
    (-4)(-3) > 0
    12 > 0
    True statement. Thus, C is correct.
    Be careful The question asks which of the following must be true.
    We found an example that makes E false so it is eliminated. the same also eliminated A , B, & D. BUT not C.
    Counterexamples eliminate statements. BUT EXAMPLES CANNOT PROVE ANYTHING.
    So to complete this question, you must explain why C must be true.
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    Re: Positive a/b

    Quote Originally Posted by Plato View Post
    Be careful The question asks which of the following must be true.
    We found an example that makes E false so it is eliminated. the same also eliminated A , B, & D. BUT not C.
    Counterexamples eliminate statements. BUT EXAMPLES CANNOT PROVE ANYTHING.
    So to complete this question, you must explain why C must be true.
    The answer is C because ab > 0 is a true statement given two negative numbers multiplied and two positive numbers multiplied.
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