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I have a problem that looks something like this:

The difference of the quotient of a number and $-2$ from $12$ is $15$.

So I started off like this:


Then I subtracted the $12$ from both sides to get:


Where would I go on from here?

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up vote 3 down vote accepted

You can multiply the numerator and denominator by $-1$ to clear the signs, then multiply by $2$ to clear the denominator.

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$$-\frac{x}{-2} = 3 \;\; \implies \;\;\not - \frac{x}{\not - 2} = 3\; \;\implies\;\;\frac{x}{2} = 3 \;\;\implies\;\; x = 2\cdot 3 = 6$$

Note that $\;-\dfrac{x}{2} = \dfrac{-x}{-2} = \dfrac{-1\cdot x}{-1\cdot 2}$ and now we can cancel $-1$ from both numerator and denomintor, or else mutltiply the numerator and denominator by $-1$; either way, it leaves us with just $\dfrac{x}{2}$.

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Hmm, thank you. However, what was done to convert the -x/-2 to x/2? – user60161 Jan 29 '13 at 23:56
@user60161: That was the multiply numerator and denominator by $-1$. Remember that minus times minus gives plus. – Ross Millikan Jan 30 '13 at 0:04
$\dfrac {-x}{-2} = \dfrac{-1\cdot -x}{-1\cdot -2} = \dfrac{x}{2}$. Multiplying by $\dfrac{-1}{-1} = 1$ does not change the value of the fraction. – amWhy Jan 30 '13 at 0:05
You can also think of it as factoring out $-1$ from both numerator and denominator, and canceling. – amWhy Jan 30 '13 at 0:06
Thank you, I understand it much better now! – user60161 Jan 30 '13 at 0:08

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