Whenever we simplify a rational expression, what we are looking for is the same thing in the numerator and the denominator if we have the same thing in both places we cancel them out to get rid of them. For this particular example what we have is 5 minus x over x minus 5, which look very similar however they are not exactly the same thing.
So what I want to do to show you this actually how is actually related is factor out a negative 1 out of one of them. It doesn’t matter which just chose one to factor out a -1. So I am going to factor out a -1 from the top.
Like I said it doesn’t matter you can do it just for the bottom as well and be just fine. So when we take that a -1 from 5 we end up with -5 and when we take out a -1 from –x, we end up with plus x and our denominator stays the same.
Doing a little bit of rearranging of this top phrase what we end up with is -1 times x minus 5 over x minus 5. So what you see is now we actually have the same thing so now we can cancel these out and leaving us with -1. These are what I call opposites I’m not exactly sure what the appropriate term is but I call them opposites because I cancel to -1.
Now what we are looking for is basically the same number where everything is the opposite sign. My 5 is positive up here my 5 is negative down here, x is positive here x is negative there. So long as everything Is opposite, you can cancel them. If we have say x minus 3 over x plus 3, there is nothing we can do with because our x’s are actually the same sign. Our 3’s are opposite but our x’s aren’t, so we can’t cancel it. It only works as if they are both opposites of each other. So when cancelling anything like this if our signs are switched we know we can cancel to -1.
Last thing I want to do with this example is talk about the domain. The domain is the values of x so we can plug in keeping in mind that we can’t divide by 0. So we always go to our original equation when we are finding the domain. If we went to a simplified version of -1 we can plug in anything but we need to go to our original statement which is 5 minus x over x minus 5. We can make the denominator 0 so x cannot be equal to 5
So that is how we deal with rational expressions when we are including opposites that cancel to -1.