Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Solving Rational Equations

After studying this lesson, you will be able to:

  • Solve rational equations.

To solve an equation with fractions (rational equations), multiply each term by a common denominator. This will eliminate the fractions. Then, solve the equation that remains.


Example 1

The common denominator is 6x so we multiply each term by 6x:

Now, we have 3 multiplications to do - this will eliminate the variables
Collect like terms
29 = x  

**remember, the solution cannot cause any denominator to equal zero - if it does, it is an extraneous solution


Example 2

The common denominator is 60 so we multiply each term by 60:

Now, we have 3 multiplications to do - this will eliminate the variables
30 - 15r + 36r = 20r + 40 Collect like terms
30 + 21r = 20r + 40 Get the variables together by subtracting 20r from each side
30 + r = 40 Subtract 30 from each side
r = 10  


Example 3

To find the common denominator we have to factor the second denominator:

The common denominator is (m + 1) (m - 1) so we multiply each term by (m + 1) (m - 1):

Now, we have 3 multiplications to do - cancel out identical binomials

-2m ( m + 1 ) + (m + 3 ) = m 2 - 1 Multiply
-2m 2 - 2m + m + 3 = m 2 - 1 Collect like terms.
-2m 2 - m + 3 = m 2 - 1 This is a quadratic equation so we need to set it equal to zero. Let's move all terms to the right side.
0 = 3m 2 + m - 4  

Now, factor and set each factor equal to zero and solve.

0 = (3m + 4) ( m - 1 )

3m + 4 = 0 m - 1 = 0
m = 1

m = 1 is an extraneous solution because it will cause a denominator to equal zero.

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