I don't understand this step in turning this negative into a positive

[Nathi ORea]

Homework Statement

I don't understand the final step of this working out

Relevant Equations

I don't think there is any.

So I have this question.
I get all the working out, but then I feel like the answer should be;
-2/(b-a).
Then I thought -2/(-a+b) must just be the same thing... all good so far..
Then they somehow just to 2/(a-b) as the final answer.. I'm lost there. How does that conversion happen?
Appreciate any help.

 

[Drakkith]

Then they somehow just to 2/(a-b) as the final answer.. I'm lost there. How does that conversion happen?

I believe they just multiplied by $\frac{-1}{-1}$ to get rid of the negative on top. Since $\frac{-1}{-1}$ is equal to $1$ you aren't changing the value of the fraction when you do so.

 

[PeroK]

Homework Statement:: I don't understand the final step of this working out
Relevant Equations:: I don't think there is any.

So I have this question.
I get all the working out, but then I feel like the answer should be;
-2/(b-a).
Then I thought -2/(-a+b) must just be the same thing... all good so far..
Then they somehow just to 2/(a-b) as the final answer.. I'm lost there. How does that conversion happen?
Appreciate any help.View attachment 323312

That's a somewhat odd approach, IMO. I would have done:
$$\frac{1}{a-b} - \frac 1 {b-a} = \frac 1 {a -b} + \frac 1 {a -b} = \frac 2 {a-b}$$

 

[PeroK]

In the original solution, note that:
$$\frac{-X}{-Y} = \frac X Y$$For any expressions $X$ and $Y$.

 

[Nathi ORea]

I believe they just multiplied by $\frac{-1}{-1}$ to get rid of the negative on top. Since $\frac{-1}{-1}$ is equal to $1$ you aren't changing the value of the fraction when you do so.

Ok.. that’s what they did aye

Is there a reason why they did that?

 

[Nathi ORea]

Ok.. that’s what they did aye

Is there a reason why they did that?

Is it just neater to have a positive as the numerator or something?

 

[Drakkith]

Is it just neater to have a positive as the numerator or something?

It's because both the numerator and denominator were negative. It's much neater and easier to turn such fractions into positives. I'm sure you'd agree that $\frac{-1}{-3}$ is better to write as $\frac{1}{3}$ instead.