##
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 5.5 c: **
**
**

**
**

**
**

**
**

**
**

**
****
**

**Answer: **
**
**

**
**

**
**

**
**

**
**

**
****
**

**Simplify the equation by subtracting **1** from both sides of the equation..
**

**
**
**
**

**
**

**
**

**
**

**
**

**
**

**
****
**

**Multiply both sides of the equation by an expression that represents the
lowest common denominator. The expression **
** is
the smallest expression because it is the smallest expression that is
divisible by all the denominators.**

**
**

**
**

**
**

**
****
**

**
**

**
**

**
**
**
**

**This expression is not easily factored. You can solve this equation using
iteration or by graphing.**

**
**

**
**

**
**

**
****
**

**With iteration, choose two real numbers and evaluate **
*f*(*x*)=2*x*^{3}-21*x*^{2}+55*x*-506** at each of the two numbers. You are looking for
a sign change. If there is a sign change, then you know that one real
solution is located between the two values you choose to test.
**

**
**
**
**

**This indicates that a real solution exists between **0** and **20.** Now narrow
the range to **10** and **20.**
**

**
**

**
**
**
**

**This indicates that real solution exists between **10** and **20.** Narrow the
range again. Since **-56** is close to 0, choose a number near **10**.
**

**
**
**
**

**You know a real solution exists between **10** and **11.** Narrow the range
again.
**

**
**
**
**

**You know a real solution exists between **10** and **1.5.** Narrow the range
again.
**

**
**
**
**

**You know a real solution exists between **10** and **10.25.** Narrow the range
again.
**

**
**
**
**

**You know a real solution exists between **10.2** and **10.25.** You can keep
going until you have the accuracy you need in your answer.
**

**
**

**
**

**
**

**
**

**
**

**
**

**
**

**
**

**
**

**
**

**
**

**
**

**
****
**

**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 spot, **
**
**

**
**

**
****
**

**
**

**
**

**
**

**
**

**
**

**
**

**
**

**
****
**

**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**

*
*

*
[Algebra]
[Trigonometry]
**
*
[Geometry]
[Differential Equations]
[Calculus]
[Complex Variables]
[Matrix Algebra]
S.O.S MATHematics home page

Do you need more help? Please post your question on our
S.O.S. Mathematics CyberBoard.

Author:
Nancy Marcus

Copyright © 1999-2017 MathMedics, LLC. All rights reserved.

Contact us

Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA

users online during the last hour