How do we simplify complex fractions?

            Complex fractions can be very intimidating, but once you get used to them there a breeze. Lets take a look at one:

1

_

x

__

1

_

x+1

      Now the first step is to just rewrite the equation so it looks like a simple division problem. So it's(1/x)/(1/x+1). Now with all division problems we use Keep. Flip. Change. We keep 1 over x, flip the second half so it is x+1 over 1, and change the sign to division sign to multiplication.

      The last thing you do is cancel out what you can in this case that would be the ones. You are left with a final answer of:

x+1

____

x

       It really is that simple and straight forward. You're all done!

How do we solve rational equations?

          The single thing that complicates solving rational equations is getting that common denominator. Much like adding and subtracting rational expressions the denominators must be the same. Let's take a look at an example:
                                         
        Our common denominator is going to be 5x(x+2) so you have to multiply each part of the equation so that they each have the same denominator. Multiply the first section by 5x, the second by 5(x+2), and the third by x+2. After all of that your new equation will look like this:
                            
             At this point, the denominators are the same. So do they really matter? Not really (other than for saying what values x can't be). So cross out the denominators and you have an equation that looks like: 15x–(5x + 10) =x+2. The rest is cake we are trying to get x alone so you solve like a regular equation. First distribute. Then The rest looks like:
                                      15x – (5x + 10) = x +2
                                          10x – 10 = x + 2
                                                9x = 12 
                                            x = ¹²/₉ = ⁴/₃
                 Since x = ⁴/₃ won't cause any division-by-zero problems in the fractions in the original equation, then this solution is valid. And that's all there is to it!

How do we multiply and divide rational expressions?

          When working with regular fractions, multiplying and dividing is much easier than adding and subtracting. The sames rules apply for simplifying rational expressions. Lets take a look at an example:      

                                                     

      Our first step is to completely factor the numerator and denominator. The first numerator factors to (x-2)(x-6) and the first denominator is a perfect square so that factors into (x+4)(x-4). The second numerator factors into 4(x+4) and the second denominator factors into (x-2)(x-2). This leaves you with (x-2)(x-6)/(x+4)(x-4) times 4(x+4)/(x-2)(x-2). 

       The next step is to cancel out all the terms that are the same, like this: 

                                     

      You can only cancel out terms that are exactly the same like (x-2) on top and ONE of the (x-2)s on the bottom. One you have canceled all things the same on the top and bottom you combine what you have left as your final answer. In the numerator you were left with 4 and (x-6) and in the denominator you were left with (x-4) and (x-2). You put all of hat together like the last section shown above and you have your final answer. Much easier than it looks. 

      Just remember to can not cancel TERMS only FACTORS. Meaning if we go back to our original equation you can not just cancel x²   in the numerator and denominator. You must factor first.

      When dividing rational expressions you do the same exact thing you do when multiplying. The key to dividing is to remember Keep. Flip. Change. Keep the first set the same, flip the second set, and change the operation to multiplication. Then you factor and cancel out like you normally would.