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Math Help - Solving inequality by multiplying across

  1. #1
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    Solving inequality by multiplying across

    Just looking for some guidance on the following problem:

    Find the least value of n (Natural number) for which (5n + 3 / 7 - n) < 1, n <> 7

    One approach:

    5n + 3 < 7 - n
    6n + 3 < 7
    6n < 4
    3n < 2
    n < 2/3

    Since n is a Natural number we get n = 0 as the least value

    The 1st step above involves multiplying across by 7 - n, however 7 - n is negative for n > 7 so this would equate to multiplying an inequality by a negative number in which case the sign would have to be changed.

    Is it valid to multiply across by 7 - n as described?

    Thanks
    Corbomite1
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  2. #2
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    Re: Solving inequality by multiplying across

    It's valid but you will then need to consider two cases. When n > 7, the denominator is negative and so will change the sign of the inequality.

    So Case 1: n < 7

    \displaystyle \begin{align*} \frac{5n + 3}{7-n} &< 1 \\ 5n+3 &< 7-n \\ 6n+3 &> 7 \\ 6n &> 4 \\ n &> \frac{2}{3}  \end{align*}

    So the values of n which satisfy this case are \displaystyle \begin{align*} n \in \left( \frac{2}{3}, 7 \right) \end{align*}


    Now consider Case 2, where n > 7...
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    Re: Solving inequality by multiplying across

    Thanks for the quick reply! I have 2 further questions:

    In case 1 above we have n < 7 which means 7 - n > 0 so we should be able to multiply across by 7 - n without changing the sign, yet you have changed the sign when arriving at 6n + 3 > 7

    In case 2, n > 7 which means 7 - n < 0 so we need to change the sign when multiplying across by 7 - n:

    5n + 3 / 7 - n < 1

    5n + 3 > 7 - n [change sign]

    6n + 3 > 7

    6n > 4

    3n > 2

    n > 2/3

    So this case cannot yield the least value of n for which the inequality holds, so we revert to n < 2/3 which gives n = 0 as n has to be a Natural number.

    Correct?

    Thanks a lot
    Corbomite1
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    Re: Solving inequality by multiplying across

    Quote Originally Posted by corbomite1 View Post
    Just looking for some guidance on the following problem:
    Find the least value of n (Natural number) for which (5n + 3 / 7 - n) < 1, n <> 7
    I much prefer to avoid the problem cases of cross multiplying. So I write it as:
    \\\frac{5n+3}{7-n}-1<0\\\frac{6n-4}{7-n}<0\\n=0\text{ or }n>7

    Be careful. Some textbooks and/or authors do not count zero among the natural numbers.
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    Re: Solving inequality by multiplying across

    Nice solution, thanks!
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    Re: Solving inequality by multiplying across

    Quote Originally Posted by corbomite1 View Post
    Thanks for the quick reply! I have 2 further questions:

    In case 1 above we have n < 7 which means 7 - n > 0 so we should be able to multiply across by 7 - n without changing the sign, yet you have changed the sign when arriving at 6n + 3 > 7
    Yes that would be a typo. I'm sure you can fix it...
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