Intermediate Algebra
Tutorial 33: Adding and Subtracting Rational Expressions

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Learning Objectives

After completing this tutorial, you should be able to:

Find the least common denominator of rational expressions.
Add and subtract rational expressions.

Introduction

Do you ever feel dazed and confused when working with fractions? If so, you are not alone. This is your lucky day! We have a whole other tutorial devoted to rational expressions (fractions). In this tutorial we will be looking at adding and subtracting them. If you need a review on simplifying, multiplying and dividing rational expressions, feel free to go back to Tutorial 32: Multiplying and Dividing Rational Expressions. It is time to get started with this tutorial.

Tutorial

Adding or Subtracting Rational Expressions
with Common Denominators

$adding fractions$

Step 1: Combine the numerators together.

Step 2: Put the sum or difference found in step 1 over the common denominator.

Step 3: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

Why do we have to have a common denominator
when we add or subtract rational expressions?????

Good question. The denominator indicates what type of fraction that you have and the numerator is counting up how many of that type you have. You can only directly combine fractions that are of the same type (have the same denominator). For example if 2 was my denominator, I would be counting up how many halves I had. If 3 was my denominator, I would be counting up how many thirds I had. But I would not be able to add a fraction with a denominator of 2 directly with a fraction that had a denominator of 3 because they are not the same type of fraction. I would have to find a common denominator first, which we will cover after the next two examples.

Example 1: Perform the indicated operation

Since the two denominators are the same, we can go right into adding these two rational expressions.

Step 1: Combine the numerators together

AND

Step 2: Put the sum or difference found in step 1 over the common denominator.

*Common denominators

*Combine the numerators
*Write over common denominator

Step 3: Reduce to lowest terms.

*Factor out a GCF of 5 in num.

*Divide out the common factor of 3x + 1

Example 2: Perform the indicated operation

Since the two denominators are the same, we can go right into subtracting these two rational expressions.

Step 1: Combine the numerators together

AND

Step 2: Put the sum or difference found in step 1 over the common denominator.

*Subtract EVERY term of second num.

Step 3: Reduce to lowest terms.

*Factor the trinomial in num.
*No common factors

Least Common Denominator (LCD)

Step 1: Factor all the denominators

If you need a review on factoring, go to any or all of the following tutorials: Tutorial 27: The GCF and Factoring by Grouping, Tutorial 28: Factoring Trinomials, or Tutorial 29: Factoring Special Products.

Step 2: The LCD is the list of all the DIFFERENT factors in the denominators raised to the highest power that there is of each factor.

Example 3: Find the LCD of

Step 1: Factor all the denominators.

The first denominator "3x" has two factors, 3 and x.

The second denominator "2" has only one factor, 2.

Step 2: The LCD is the list of all the DIFFERENT factors in the denominators raised to the highest power that there is of each factor.

Putting all the different factors together, we get (3)(x)(2) = 6x for our LCD.

Example 4: Find the LCD of

Step 1: Factor all the denominators.

The first denominator has the following two factors:

*Factor the diff. of two squares

The second denominator has the following factor:

*Factor the perfect square trinomial

Step 2: The LCD is the list of all the DIFFERENT factors in the denominators raised to the highest power that there is of each factor.

Putting all the different factors together and using the highest exponent, we get the following LCD:

Adding and Subtracting Rational Expressions
Without a Common Denominator

Step 1: Find the LCD as shown above if needed.

Step 2: Write equivalent fractions using the LCD if needed.

If we multiply the numerator and denominator by the exact same expression it is the same as multiplying it by the number 1. If that is the case, we will have equivalent expressions when we do this.

Now the question is WHAT do we multiply top and bottom by to get what we want? We need to have the LCD, so you look to see what factor(s) are missing from the original denominator that is in the LCD. If there are any missing factors than that is what you need to multiply the numerator AND denominator by.

Step 3: Combine the rational expressions as shown above.

Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

Example 5: Perform the indicated operation

Step 1: Find the LCD as shown above if needed.

The first denominator has the following two factors:

*Factor the trinomial

The second denominator has the following factor:

Putting all the different factors together and using the highest exponent, we get the following LCD:

Step 2: Write equivalent fractions using the LCD if needed.

Since the first rational expression already has the LCD, we do not need to change this fraction.

*Rewriting denominator in factored form

Rewriting the second expression with the LCD:

*Missing the factor of (x - 1) in the den.

*Mult. top and bottom by (x - 1)

Step 3: Combine the rational expressions as shown above.

*Subtract EVERY term of ( ).

Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

This rational expression cannot be simplified down any farther. So, our answer is

Example 6: Perform the indicated operation

Step 1: Find the LCD as shown above if needed.

The first denominator has the following factor:

The second denominator has the following two factors:

*Factor the sum of cubes

Putting all the different factors together and using the highest exponent, we get the following LCD:

Step 2: Write equivalent fractions using the LCD if needed.

Rewriting the first expression with the LCD:

*Missing the trinomial factor in the den.

*Mult. top and bottom by the trinomial

Since the second rational expression already has the LCD, we do not need to change this fraction.

*Rewriting denominator in factored form

Step 3: Combine the rational expressions as shown above.

*Add the two numerators together
*Write over the LCD

Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

*Divide out the common factor of x + y

Example 7: Perform the indicated operation

Step 1: Find the LCD as shown above if needed.

The first denominator has the following factor:

The second denominator, 7 - x, looks like the first denominator except the signs are switched. We can rewrite this as

*Factor out a -1

Putting all the different factors together and using the highest exponent, we get the following LCD:

Note that I did not put the -1 that was in front of the second denominator's (x - 7) factor. In step 3, I will put the negative into the problem by placing it in the numerator of that second fraction.

Step 2: Write equivalent fractions using the LCD if needed.

Now the two fractions have a common denominator, so we do not have to rewrite the rational expressions.

Step 3: Combine the rational expressions as shown above.

*Combine the two num.
*Write over the common den.

Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

This rational expression cannot be simplified down any farther. So, our answer is

Example 8: Perform the indicated operation

Step 1: Find the LCD as shown above if needed.

The first denominator has the following factor:

The second denominator has the following two factors:

*Factor out a GCF of 2

The third denominator has the following two factors:

*Factor out a GCF of a

Putting all the different factors together and using the highest exponent, we get the following LCD:

Step 2: Write equivalent fractions using the LCD if needed.

Rewriting the first expression with the LCD:

*Missing 2(a + 2) in the den.

*Mult. top and bottom by 2(a + 2)

Rewriting the second expression with the LCD:

*Missing a in the den.

*Mult. top and bottom by a

Rewriting the third expression with the LCD:

*Missing 2 in the den.

*Mult. top and bottom by 2

Step 3: Combine the rational expressions as shown above.

*Combine the two num.
*Write over the common den.

Step 4: Reduce to lowest terms as shown in Tutorial 32: Multiplying and Dividing Rational Expressions.

*Factor out a GCF of a in the num.

*Divide out the common factors of a and (a + 2)

Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problems 1a - 1c: Perform the indicated operation.

1a.

(answer/discussion to 1a)

1b.

(answer/discussion to 1b)

1c.
(answer/discussion to 1c)

Need Extra Help on these Topics?

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

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