# Rational Expressions and Equations

## Add and Subtract Rational Expressions with Unlike Denominators


### 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.

### 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.

### 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.
Now we will add rational expressions whose denominators are monomials.
Now we are ready to tackle polynomial denominators.
The steps to use to add rational expressions are summarized in the following procedure box.
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.

### 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.
The steps to take to subtract rational expressions are listed below.
There are lots of negative signs in the next example. Be extra careful!
When one expression is not in fraction form, we can write it as a fraction with denominator 1.
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.

### Key Concepts

1. Find the Least Common Denominator of Rational Expressions


2. Add or Subtract Rational Expressions



### Practice Makes Perfect

In the following exercises, find the LCD.
In the following exercises, write as equivalent rational expressions with the given 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


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?