in the answer 2(4x-3)/3 -4x what make this unique?

Question

When I saw my answer I noticed that the numerator and the denominator were in reverse order, can anything be done to this expression?

Answer 1

Hi, Debbie.

If I understand your question correctly, your problem looks like this:

2 (4x - 3)
  3 - 4x 

What I tell students is to rewrite the denominator in descending order (of exponents) so that the problem looks like this:

2 (4x - 3)
 -4x + 3

Now, pull out a GCF of -1 from the denominator's expression:

  2 (4x - 3)
 -1 (4x - 3)

See how the 4x - 3 factors are the same and can "cancel" with each other or reduce down to 1?
This leaves us with only the 2 over -1.  
Reducing that fraction gives us our answer of -2.

Reducing polynomial fractions is all about finding factors in common between the numerator and the denominator.  Rewriting factors in descending order helps us to see that when expressions are identical over subtraction, except for their order, that they reduce to -1.  Some teachers use that as a short cut in solving the problem.  Let me show you:

Same problem:

2 (4x - 3)
  3 - 4x

From the get-go, the 4x - 3 is visibly "backwards" over subtraction to 3 - 4x.  
So cancel them out with each other, and write -1 in one of their spots.  This leaves us with 2 times -1 (or 2 div by -1, depending on where you wrote the -1).  
The answer is much more quickly obtained as -2.

Hope this helps you see it the "long" way and "short cut" way.

 

 

Answer 2

From what you wrote, I understand that the 3 - 4x is all in the denominator

To simplify this expression, multiply the fraction by -1/(-1)

2(4x - 3)/(3 - 4x) * [(-1)/(-1)]

-2(4x - 3)/[-1(3 - 4x)]

After distributing the negative in the denominator, we have

-2(4x - 3)/(-3 + 4x)

If we change the order of the terms in the denominator, we have

-2(4x - 3)/(4x - 3)

The fraction can be reduced to -2.

As you assumed, anytime the numerator and denominator have the same terms, but opposite signs, the fraction can be reduced.  You just have to multiply by -1.