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 in solving a rational equation is to convert the equation to an equivalent equation without denominators.

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

• Remember that you are trying to isolate the variable.

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

Solve for x in the following equation.

Example 1:

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 is . If either 3 or-3 turn out to be the solutions, you must discard them as extraneous solutions.

Rewrite the problem so that every denominator is factored

Multiply both sides by the least common (x-3)(x+3) multiple (the smallest expression that all the denominators will divide into evenly).

which is equivalent to

which can be rewritten

which can be rewritten

which can be rewritten

The answer is

Check this answer in the original equation.

Check the solution x=5 by substituting 5 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 equal to the right side of the original equation after we substitute the value 5 for x, then x=5 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 5. This means that the real solution is 5.

If you would like to work another example, click on Example

If you would like to test yourself by working some problems similar to this example, click on Problem

If you would like to go back to the equation table of contents, click on Contents

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