How Do You Simplify Complex Fractions?

[Gringo123]

Can anyone explain to me what approach I need to take to solve this type of problem?

Express x/3x+1 - 2/8x-1

 

[tiny-tim]

Hi Gringo123!

Put (3x+1)(8x-1) on the bottom.

 

[Gringo123]

Thanks Tim
The answer that I have come up with is:
(8x - 1) (x + 1) / 3x + 1) (8x - 1)
Is that correct? If it is, can I not cancel out the (8x - 1) from the top and bottom, leaving me with just:
x + 1 / 3x - 1
?

 

[Gringo123]

and then surely I can cancel x + 1 / 3x - 1 further to just x / 3x?

 

[Mark44]

Thanks Tim
The answer that I have come up with is:
(8x - 1) (x + 1) / 3x + 1) (8x - 1)

Is that correct?

No. Show us what you got before you factored the numerator.

If it is, can I not cancel out the (8x - 1) from the top and bottom, leaving me with just:
x + 1 / 3x - 1
?

If what you got previously was correct, then yes, this would be valid. However, instead of writing x + 1 / 3x - 1, you need parentheses around the numerator and denominator, like this (x + 1) / (3x - 1). Note that this isn't the right answer.

 

[Mark44]

and then surely I can cancel x + 1 / 3x - 1 further to just x / 3x?

Surely not, and you have made at least two errors coming up with this. Cancelling means removing factors that are the same in numerator and denominator.

 

[Gringo123]

This is how I arrived at x + 1 / 3x + 1:

* x/3x + 1 - 2/ 8x-1
* x(8x - 1) - 2(3x + 1) / (3x + 1) (8x - 1)
* 8x squared - x - 6x - 1 / (3x + 1) (8x - 1)
* 8x squared - 7x - 1 / (3x + 1) (8x - 1)
* (8x - 1) (x + 1) / (3x + 1) (8x - 1)
* (x + 1) / (3x + 1)

 

[Mark44]

This is how I arrived at x + 1 / 3x + 1:

* x/3x + 1 - 2/ 8x-1

Use parentheses. The above should be written as x/(3x + 1) - 2/(8x - 1). If a numerator or denominator has more than one term, you need to surround it with parentheses.

* x(8x - 1) - 2(3x + 1) / (3x + 1) (8x - 1)

Use = to indicate that expressions are equal. Except for the first line, everywhere you have used * you should have =.
The entire numerator needs parentheses around it.

* 8x squared - x - 6x - 1 / (3x + 1) (8x - 1)

You skipped a step in the above, and one of your terms in the numerator is wrong.
Use the X² button to enter exponents. You've been coming here long enough that you should start getting the hang of what features are available. If you don't see the X² button, click the Go Advanced button just below the text entry area.

* 8x squared - 7x - 1 / (3x + 1) (8x - 1)
* (8x - 1) (x + 1) / (3x + 1) (8x - 1)

Aside from the fact that 8x² - 7x - 1 is incorrect, the factorization is not (8x - 1)(x + 1). As problems get more involved, check your work in intermediate steps. Does (8x - 1)(x + 1) multiply to 8x² - 7x - 1?

* (x + 1) / (3x + 1)

 

[Mark44]

Can anyone explain to me what approach I need to take to solve this type of problem?

Express x/3x+1 - 2/8x-1

The approach is the same as for adding two fractions, such as 3/8 + 1/6.

3/8 + 1/6
$$=\frac{3}{8}*\frac{3}{3} + \frac{1}{6}*\frac{4}{4}$$
$$=\frac{9}{24} + \frac{4}{24} = \frac{13}{24}$$
The least common denominator fro 8 and 6 is 24. The first fraction is multiplied by 1 in the form of 3/3 so as to get the common denominator. The second fraction is multiplied by 1 in the form of 4/4 for the same reason.

The approach for your algebra problem is nearly identical: find the least common denominator, and then multiply each rational expression by 1 to get a common denominator for both rational expressions.