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Thread: Simplifying the rational expressions-Help!

  1. #1
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    Simplifying the rational expressions-Help!

    (t+7)(t-9)/(t+3)(t+7)*(t+3)(t+13)/(9-t)(t-13)




    and 3m= -9/(m+4)



    Thank you!
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  2. #2
    Member Jonboy's Avatar
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    $\displaystyle \frac{(t+7)(t-9)}{(t+3)(t+7)}\cdot \frac{(t+3)(t+13)}{(9-t)(t-13)}$
    look to cancel things. start off just from the top and bottom of the two fractions.

    you'll see: $\displaystyle \frac{\rlap{--------}(t+7)(t-9)}{(t+3)\rlap{--------}(t+7)}\cdot \frac{(t+3)(t+13)}{(9-t)(t-13)}$

    so now you're left with: $\displaystyle \frac{(t-9)}{(t+3)}\cdot \frac{(t+3)(t+13)}{(9-t)(t-13)}$

    now you can do some cross canceling: $\displaystyle \frac{(t-9)}{\rlap{--------}(t+3)}\cdot \frac{(\rlap{--------}t+3)(t+13)}{(9-t)(t-13)}$

    so you're left with: $\displaystyle \frac{(t-9)(t+13)}{(9-t)(t-13)}$

    no more canceling you can do. so just multiply out the top and bottom of the fraction.

    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~

    $\displaystyle 3m = - \frac{9}{(m + 4)}$
    Get rid of that fraction on the right.

    Just simply multiply by $\displaystyle m + 4$: $\displaystyle (3m)(m + 4) = - 9$

    Now just multiply the left hand side and get it equal to zero.

    Then you're solving for m like a normal quadratic problem.

    Keep in mind that m cannon be - 4 as it would make the bottom of the fraction zero in the original problem.
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  3. #3
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    Hello, ep78!

    It's already factored for you . . . Can't you cancel ??


    $\displaystyle \frac{(t+7)(t-9)}{(t+3)(t+7)}\cdot\frac{(t+3)(t+13)}{(9-t)(t-13)}$
    In the bottom of the second fravtion, we factor: .$\displaystyle 9 - t \:=\:-(t-9)$

    We have: . $\displaystyle \frac{(t+7)((t-9)}{(t+3)(t+7)} \cdot\frac{(t+3)(t+13)}{-(t-9)(t-13)} $

    And reduce: .$\displaystyle \frac{{\color{red}\rlap{/////}}(t+7){\color{blue}\rlap{/////}}(t-9)}{{\color{green}\rlap{/////}}(t+3){\color{red}\rlap{/////}}(t+7)} \cdot\frac{{\color{green}\rlap{/////}}(t+3)(t+13)}{-{\color{blue}\rlap{/////}}(t-9)(t-13)} \;\;=\;\;-\frac{t+13}{t-13} $

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  4. #4
    Member Jonboy's Avatar
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    I didn't see that Soroban, thanks for giving a more accurate answer.
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  5. #5
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    Hello, Jonboy!

    It took me an embarrassingly long time to catch the relationship
    . . between $\displaystyle a - b$ and $\displaystyle b - a$.

    Now (many years later), it's like a flashing red light.
    . . Just one more thing to watch for . . .

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  6. #6
    Member Jonboy's Avatar
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    Glad I'm not the only one.
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