Rational equation what is

Some rational expressions have a variable in the denominator. When this is the case, there is an extra step in solving them. Since division by 0 is undefined, you must exclude values of the variable that would result in a denominator of 0. These values are called excluded values. Since the denominator of each expression in the equation is the same, the numerators must be equal.

Set the numerators equal to one another and solve for x. In the following video we present an example of solving a rational equation with variables in the denominator. These types of answers are called extraneous solutions.

Set the numerators equal to one another and solve for m. Sometimes, solving a rational equation results in a quadratic. How To: Given a rational equation, solve it. Factor all denominators in the equation.

Find and exclude values that set each denominator equal to zero. Find the LCD. Multiply the whole equation by the LCD. If the LCD is correct, there will be no denominators left. Solve the remaining equation. Make sure to check solutions back in the original equations to avoid a solution producing zero in a denominator.

Solution First find the common denominator. However, for this problem, we can cross-multiply. This equation has subtracted fractions on the left-hand side, so I can't cross-multiply. Also, there's the new wrinkle of variables in the denominator. This means that I'll need to keep track of the values of x that would cause division by zero. These values cannot be part of my final answer. In this case, the denominators tell me that my answer will have the following restriction:.

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, due to division-by-zero issues.

At this point, the two sides of the equation will be equal as long as the numerators are equal. That is, all I really need to do now is solve the numerators:. Method 2: The other method is to find the common denominator but, rather than converting everything to that denominator, I'll take advantage of the fact that I have an equation here.

That is, I'll multiply through on both sides by that common denominator. This will get rid of the denominators. I've used colors below to highlight the parts that cancel off:. I view Method 2 as being quicker and easier, but this is only my personal preference. Step 4: Check for extraneous solutions. Therefore it is not included in the solution set. If this process produces a solution that happens to be a restriction, always disregard it as a solution.

Sometimes all potential solutions are extraneous, in which case we say that there is no solution to the original equation. In the next two examples, we demonstrate two ways in which rational equation can have no solutions. The equation is a contradiction and thus has no solution. It is important to point out that this technique for clearing algebraic fractions only works for equations.

Do not try to clear algebraic fractions when simplifying expressions. As a reminder, an example of each is provided below. Expressions are to be simplified and equations are to be solved.