This week in math we learned how to solve rational equations. A rational equation is like a rational expression, but in an equation (so there’s an equal sign) with most-likely multiple rational expressions. Because it’s an equation and has an equal sign, you can solve it and find solution(s). The first step is to factor any parts of the equation if possible. If not, move on to the next step, which is to find the non-permissible values, which we previously learned about.
Now, we can begin solving. There are multiple ways you can solve a rational equation, I’ll talk about a few different ways. Depending on what the equation looks like, one way of solving might be easier than others. If there are two rational expressions in the equation that have a common denominator, then you can combine them (collect like terms). To note, you may have to move one of the expressions to the other side of the equation to combine both expressions. After this, you can isolate x using simple algebra.
Another way of solving is to cross multiply. This works if there is one rational expression on each side of the equal sign. So, there can’t be any other numbers in the equation or expressions, only two rational expressions. Move any expressions you have to, then you can cross multiply. So, multiply the nominator on one side by the denominator on the other side for both sides of the equation. From here, use algebra to find the solutions.
The last way to solve that I’ll talk about is multiplying through with a common denominator. Find the common denominator of all the numbers/expressions in the equation and multiply each part of the equation by the common denominator. Cancel out any numbers possible, which will get rid of the denominators, therefore the fractions as well. From here you can collect like terms and use algebra to solve. I’ve included an example of these three ways of solving below.