Solving a Rational Equation - Concept
Solving rational equations is substantially easier with like denominators. When solving rational equations, first multiply every term in the equation by the common denominator so the equation is "cleared" of fractions. Next, use an appropriate technique for solving for the variable.
So the easiest way to solve a rational equation is probably what you remember from Algebra 1 which is just basicaly cross multiplying both sides so you do as you take one denominator multiply it by the numerator the other side and vice versa and solve it out okay so that's great if we have a single fraction equal to a single fraction okay but that's not always going to be the case sometimes you're going to be adding and subtracting things on one side and getting those together to get one fraction is isn't always going to be the easiest so there is another way to solve stuff like this and basically what we want to do is multiply everything by the least common denominator. And what that does is really get rid of our fractions all together.
Okay, the first thing you want to do only when you're dealing with that is to factor our denominator so we can find what our least common denominator is so we factor this out this is x+1, x-1 and then we multiply both sides by the least common denominator. The least common denominator in this case is x+1 times x-1 okay so multiply everything by this when we distribute this into this first term, the x+1's cancel and we're just left with 3 times x-1. Over here everything cancels because x+1, x-1 is the same thing as x squared minus 1 so we end up with this is equal to 1 and now we have a equation without any denominator to solve, so this is just 3x-3 is equal to 1 so I open this up I add 3, 3x is equal to 4 divide by 3x is equal to four thirds okay?
The one thing we do have to be careful with when solving rational equations is our domain issues okay and by domain issues are those that the values of x that can put in so you always want to look at your beginning statement and see what restrictions you have on x so here I can't have x is equal to 1 or -1 because I'd be dividing by 0 and I can't do that, so whenever you get your answer you always want to make sure that it's in the domain of the statement you started with. Here x is 4 thirds that's not a problem because we said the only issues we had were 1 and -1 okay so solving a rational equation by multiplying by the least common denominator, it's really cool everything cancels out the denominator and we're able to solve a equation in this case it's a linear equation sometimes you'll end up with quadratics but either way we're going to end up with something we know how to solve.