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I was wondering that is

$x=-\frac{m}{n}$

same as

$-x=\frac{m}{n}$

The question popped into my mind when had

$x=-\frac{11}{14}$ or $-x=\frac{11}{14}$

as an anwser to one of my equations. Was the $x$ positive or negative $-x$ only depended on which side I putted numbers and X's in my calculations.

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  • $\begingroup$ When solving an equation, you're usually expected to denote the value of $x$, not the value of $-x$. $\endgroup$ – barak manos Jan 14 '16 at 10:22
  • $\begingroup$ If you want to quote the value of $x$ then you would write $x = -11/15$ (which is negative). By writing $-x = 11/15$ then you are saying that (the different number) $-x$ equals to $11/15$ (which is positive). $\endgroup$ – Winther Jan 14 '16 at 10:27
  • $\begingroup$ Try Googling "multiplicative reflexive axiom". Reason as follows $$x = - \frac{m}{n}\implies cx= c(-\frac{m}{n})$$ Now let $c = -1$ $\endgroup$ – John Joy Jan 14 '16 at 14:32
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In both cases the sign of $x$ is the same. If $\frac{m}{n}>0$ Then $$x = -\frac{m}{n}<0 \Rightarrow x<0$$ In the second equation $-x = \frac{m}{n}$ we also have that $x<0$ since $$-x = \frac{m}{n} \Rightarrow x = -\frac{m}{n}$$ The same can be done when $\frac{m}{n}<0$.

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They are the same in the sense that those equations are equivalent, i.e. they have the same solutions.

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