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Math Help - How are these two equivalent

  1. #1
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    How are these two equivalent

    Hi,

    I'm not seeing how these two are equivalent: -\frac{1}{x-1} \equiv \frac{1}{1-x}.

    Could somebody show/tell me, please?
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  2. #2
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    We have a\cdot(-b)=(-a)\cdot b=-ab. In particular, when b = 1/c, we have \displaystyle\frac{a}{-c}=\frac{-a}{c}=-\frac{a}{c}. In this case, -(x-1)=1-x.

    Also, these two fractions are equal, not equivalent.
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  3. #3
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    Or (more the same) just multiply both the numerator and the denominator by negative one:

    \displaystyle -\frac{1}{x-1} = \frac{-1}{x-1} = \frac{\left(-1\right)\left(-1\right)}{\left(-1\right)\left(x-1\right)} = \frac{1}{-x+1} = \frac{1}{1-x}
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  4. #4
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    x = 3

    -(1 / (3 - 1)) = -(1 / 2) = -1/2

    1 / (1 - 3) = 1 / -2 = -1/2
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