**
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 5.3d:**

**Answer:***x*=7

**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 is
If
, or
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

The only way the product of two expressions equals zero if at least one of
the expressions is zero.

To determine what values of x will cause the expression to equal zero, we use the Quadratic Formula

Since you cannot compute the square root of a negative number in the real number system, these two solutions are not real.

The only real solution is

Check the solution *x*=7 by substituting 7 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:

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 one place: *x*=7.

This verifies our solution graphically.

We have verified our solution both algebraically and graphically.

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