Rational Expressions and Equations
65 Add and Subtract Rational Expressions with Unlike Denominators
Learning Objectives
By the end of this section, you will be able to:
 Find the least common denominator of rational expressions
 Find equivalent rational expressions
 Add rational expressions with different denominators
 Subtract rational expressions with different denominators
Before you get started, take this readiness quiz.
If you miss a problem, go back to the section listed and review the material.
Find the Least Common Denominator of Rational Expressions
When we add or subtract rational expressions with unlike denominators we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.
Let’s look at the example from Foundations. Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator.
To find the LCD of 12 and 18, we factored each number into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.
We do the same thing for rational expressions. However, we leave the LCD in factored form.
 Factor each expression completely.
 List the factors of each expression. Match factors vertically when possible.
 Bring down the columns.
 Multiply the factors.
Remember, we always exclude values that would make the denominator zero. What values of x should we exclude in this next example?
Find the LCD for .
Find the LCD for .
Find the LCD for .
Find Equivalent Rational Expressions
When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD.
We will do the same thing for rational expressions.
Rewrite as equivalent rational expressions with denominator :
Factor each denominator.  
Find the LCD.  
Multiply each denominator by the ‘missing’ factor and multiply each numerator by the same factor.  
Simplify the numerators. 
Rewrite as equivalent rational expressions with denominator :
Rewrite as equivalent rational expressions with denominator :
Add Rational Expressions with Different Denominators
Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.
Add:
Find the LCD of 12 and 18.  
Rewrite each fraction as an equivalent fraction with the LCD.  
Add the fractions.  
The fraction cannot be simplified. 
Add:
Add:
Now we will add rational expressions whose denominators are monomials.
Add:
Find the LCD of and .  
Rewrite each rational expression as an equivalent fraction with the LCD.  
Simplify.  
Add the rational expressions.  
There are no factors common to the numerator and denominator. The fraction cannot be simplified. 
Add:
Add:
Now we are ready to tackle polynomial denominators.
Add:
Add:
Add:
The steps to use to add rational expressions are summarized in the following procedure box.
 Determine if the expressions have a common denominator.
Yes – go to step 2.
No – Rewrite each rational expression with the LCD.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.  Add the rational expressions.
 Simplify, if possible.
Add:
Do the expressions have a common denominator? No. Rewrite each expression with the LCD. 

Find the LCD.  
Rewrite each rational expression as an equivalent rational expression with the LCD.  
Simplify the numerators.  
Add the rational expressions.  
Simplify the numerator.  
Factor the numerator.  
There are no factors common to the numerator and denominator. The fraction cannot be simplified. 
Add:
Add:
Avoid the temptation to simplify too soon! In the example above, we must leave the first rational expression as to be able to add it to . Simplify only after you have combined the numerators.
Add:
Do the expressions have a common denominator? No. Rewrite each expression with the LCD. 

Find the LCD.  
Rewrite each rational expression as an equivalent fraction with the LCD.  
Simplify the numerators.  
Add the rational expressions.  
Simplify the numerator.  
The numerator is prime, so there are no common factors. 
Add:
Add:
Subtract Rational Expressions with Different Denominators
The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.
Subtract:
Subtract:
Subtract:
The steps to take to subtract rational expressions are listed below.
 Determine if they have a common denominator.
Yes – go to step 2.
No – Rewrite each rational expression with the LCD.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.  Subtract the rational expressions.
 Simplify, if possible.
Subtract:
Do the expressions have a common denominator? No. Rewrite each expression with the LCD. 

Find the LCD.  
Rewrite each rational expression as an equivalent rational expression with the LCD.  
Simplify the numerators.  
Subtract the rational expressions.  
Simplify the numerators.  
Factor the numerator to look for common factors.  
Remove common factors.  
Simplify. 
Subtract:
Subtract:
There are lots of negative signs in the next example. Be extra careful!
Subtract:
Factor the denominator.  
Since and are opposites, we will mutliply the second rational expression by.  
Simplify.  
Do the expressions have a common denominator? No.  
Find the LCD.  
Rewrite each rational expression as an equivalent rational expression with the LCD.  
Simplify the numerators.  
Simplify the rational expressions.  
Somplify the numerator.  
Factor the numerator to look for common factors.  
Simplify. 
Subtract:
Subtract:
When one expression is not in fraction form, we can write it as a fraction with denominator 1.
Subtract:
Write as to have 2 rational expressions.  
Do the rational expressions have a common denominator? No.  
Find the LCD of and LCD =  
Rewrite as an equivalent rational expression with the LCD.  
Simplify.  
Subtract the rational expressions.  
Simplify.  
Factor to check for common factors.  
There are no common factors; the rational expression is simplified. 
Subtract:
Subtract:
 Determine if the expressions have a common denominator.
Yes – go to step 2.
No – Rewrite each rational expression with the LCD.
Find the LCD.
Rewrite each rational expression as an equivalent rational expression with the LCD.  Add or subtract the rational expressions.
 Simplify, if possible.
We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example we will start by factoring all three denominators to find their LCD.
Simplify:
Do the rational expressions have a common denominator? No.  
Find the LCD.  
Rewrite each rational expression as an equivalent rational expression with the LCD.  
Write as one rational expression.  
Simplify.  
Factor the numerator, and remove common factors.  
Simplify. 
Simplify:
Simplify:
Key Concepts
 Find the Least Common Denominator of Rational Expressions
 Factor each expression completely.
 List the factors of each expression. Match factors vertically when possible.
 Bring down the columns.
 Multiply the factors.
 Add or Subtract Rational Expressions
 Determine if the expressions have a common denominator.
Yes – go to step 2.
No – Rewrite each rational expression with the LCD. Find the LCD.
 Rewrite each rational expression as an equivalent rational expression with the LCD.
 Add or subtract the rational expressions.
 Simplify, if possible.
 Determine if the expressions have a common denominator.
Practice Makes Perfect
In the following exercises, find the LCD.
In the following exercises, write as equivalent rational expressions with the given LCD.
LCD
LCD
LCD
LCD
LCD
LCD
LCD
LCD
In the following exercises, add.
In the following exercises, subtract.
In the following exercises, add and subtract.
In the following exercises, simplify.
Everyday Math
Decorating cupcakes Victoria can decorate an order of cupcakes for a wedding in hours, so in 1 hour she can decorate of the cupcakes. It would take her sister 3 hours longer to decorate the same order of cupcakes, so in 1 hour she can decorate of the cupcakes.
 ⓐ Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions
 ⓑ Evaluate your answer to part (a) when
Kayaking When Trina kayaks upriver, it takes her hours to go 5 miles, where is the speed of the river current. It takes her hours to kayak 5 miles down the river.
 ⓐ Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding
 ⓑ Evaluate your answer to part (a) when to find the number of hours it would take Trina if the speed of the river current is 1 mile per hour.
Writing Exercises
Felipe thinks is
 ⓐ Choose numerical values for x and y and evaluate
 ⓑ Evaluate for the same values of x and y you used in part (a).
 ⓒ Explain why Felipe is wrong.
 ⓓ Find the correct expression for
Answers may vary.
Simplify the expression and explain all your steps.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 110, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?