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I have been trying to simplify a rational expression; however, I am not able to reach the simplest form of the expression, here is my work:

$$\frac{5g+2}{5g-2}\div\frac{3g-1}{1-3g}$$

Then I try to simplify the equation by first multiplying by its reciprocal

$$\frac{5g+2}{5g-2}\times\frac{1-3g}{3g-1}$$

So once I have multiplication I find a common denominator

$$\frac{(5g+2)(3g-1)}{(5g-2)(3g-1)}\times\frac{(1-3g)(5g-2)}{(3g-1)(5g-2)}$$

However, after I cross out the terms due to division I end up with the same expression at the start

$$\frac{5g+2}{5g-2}\div\frac{3g-1}{1-3g}$$

What step am I missing to simplify it even more?

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    I noticed that $$ \frac{3g-1}{1-3g} = -1. $$ Is that helpful? – 2017-01-12
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    Could you explain how? – 2017-01-12
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    Because $3g - 1 = - (1 - 3g) = (-1) (1 - 3g)$. Basically, the numerator and denominator of your 2nd expression are negatives of each other. – 2017-01-12
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    You don't look for a common denominator when multiplying only when adding/substracting. – 2017-01-12

2 Answers 2

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One might wish to note, as mentioned in the comments by @avs, that

$$\frac{3g-1}{1-3g}=\frac{-(1-3g)}{1-3g}=-1$$

Thus, we have

$$\frac{5g+2}{5g-2}\times(-1)$$

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    I see it, I am so blind! thanks! – 2017-01-12
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    @pablo No problem! Your welcome :-P – 2017-01-12
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    Thanks for the acknowledgement to avs. :-) – 2017-01-12
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    @amWhy :-) Thanks for reminding me to put it in. – 2017-01-12
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.The step you are missing is that $-1(1-3g) = -1 + 3g = 3g-1$, therefore $\frac{3g-1}{1-3g} = -1$! Thank you to avs for the point out in the comments.

Therefore, $$ \frac{5g+2}{5g-2} \div \frac{3g-1}{1-3g} = \frac{5g+2}{5g-2} \div -1 = \frac{-1(5g+2)}{5g-2} = \frac{-5g-2}{5g-2} $$

You can check that: $$ \frac{-5g-2}{5g-2} = \frac{-4}{5g-2} -1 $$

This simplification puts the expression in "partial sum" form, where only constants are on the numerators.

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    "The step you are missing is that $-1(1-3g) = -1 + 3g = 3g-1$, therefore $\frac{3g-1}{1-3g} = -1$!" **As mentioned in a comment from @avs** – 2017-01-12
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    @amWhy Slow internet. It took me some time to load my answer, when none of the other answers (or comments) had been loaded. I am seeing your comment late as well. I added a further simplification, so my answer gives more "detail" than the comment, for somebody else's help. I would like to ask, how do you write in bold in the comments, because that's not there in the tutorials? – 2017-01-12
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    You use \*\* bold text here \*\*. – 2017-01-12
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    **Thank you** @SimpleArt – 2017-01-12
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    Actually bold text does't allow spaces between the **.** and the text to boldface. `**bold text**` will yield **bold text**, whereas `** bold text **` will yield ** bold text ** – 2017-01-12
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    @amWhy Thank you for the comment. – 2017-01-12