## EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)

Note:

• A rational equation is an equation where at least one denominator contains a variable.

• When a denominator contains a variable, there is a restriction on the domain. The variable cannot take on any number that would cause any denominator to be zero.

• The first step is solving a rational equation is to convert the equation to an equation without denominators. This new equation may be equivalent (same solutions as the original equation) or it may not be equivalent (extraneous solutions).

• The next step is to set the equation equal to zero and solve.

• Remember that you are trying to isolate the variable.

• Depending on the problem, there are several methods available to help you solve the problem.

If you would like an in-depth review of fractions, click on Fractions.

Solve for x in the following equation.

Problem 5.2h:

Comment on answers: You may wonder why we give you the answers in two forms: exact and approximate. There is a reason. Students seem perplexed when they think they have worked a problem correctly and yet, their exact answers differ from the exact answers in the book. The student is not necessarily wrong. Depending on the method chosen to work the problem, exact answers have a different look. How do you know whether your exact answer is equivalent to a different looking exact answer in the book? Simplify both. If both exact answers are correct, they will both simplify to the same approximate answer.

Next time your answer differs from the answer in the book, simplify both. If the approximate answers are the same, you are correct. If not, go back to the drawing board and try to find your mistake.

Solution:

Recall that you cannot divide by zero. Therefore, the first fraction is valid if , the second fraction is valid if , and the third fraction is valid if If either or or0 turn out to be the solutions, you must discard them as extraneous solutions.

Multiply both sides by the least common multiple (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators.

which is equivalent to

which can be rewritten as

which can be rewritten as

which can be simplified to

> Solve for x using the quadratic formula

The answers are However, this may or may not be the answer. You must check the solution with the original equation.

Check the solution by substituting 3.40858123075 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is reasonably close to the right side of the original equation after we substitute the value 3.40858123075 for x, then x=3.40858123075 is a solution.

Check the solution by substituting 0.765331812733 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is reasonably close to the right side of the original equation after we substitute the value 0.765331812733 for x, then x=0.765331812733 is a solution.

You can also check your answer by graphing (formed by subtracting the right side of the original equation from the left side). Look to see where the graph crosses the x-axis; that will be the real solution. Note that the graph crosses the x-axis at two places: .

We have verified the solution two ways.

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Author: Nancy Marcus