Big Ideas Math Algebra 2, 2014
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Big Ideas Math Algebra 2, 2014 View details
4. Adding and Subtracting Rational Expressions
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Exercise 2 Page 388

It is not the same process to add or subtract fractions with like denominators as it is to add or subtract fractions with unlike denominators.

See solution.

Practice makes perfect

Adding and subtracting rational expressions is done using the exact same thought process as adding and subtracting numerical fractions.

Adding or Subtracting With Like Denominators

If two numerical fractions or rational expressions have like denominators, we only need to add or subtract the numerators. Consider the following examples of addition.
Note that in both cases the denominators are the same. Therefore, to simplify the expressions, we need to add the numerators. Let's start with the numerical fractions.
Let's now proceed in the same way to add the rational expressions.

Adding or Subtracting With Unlike Denominators

There are two ways of adding and subtracting rational expressions and fractions with unlike denominators. Let's explore them one at a time.

First Method

The first method consists of multiplying and dividing each term by the denominator of the other term. This will result in like denominators. Therefore, from here, we can add or subtract the numerators.
This method is valid for both numerical fractions and rational expressions. Consider the following examples of subtraction.
In both cases, we have unlike denominators. Let's use the method we explained above to perform the subtraction. We will start with the numerical fractions.
Let's now proceed in the same way to subtract the rational expressions.
Multiply parentheses
In this case, we can factor both the numerator and the denominator. Let's do it and cancel out any possible common factors.

Second Method

The second method for adding and subtracting numerical fractions or rational expressions with unlike denominators relies on multiplying all numerators and denominators by the least common denominator. We will consider the same examples as before and subtract using this method.
Let's start by subtracting the numerical fractions. Note that is a multiple of This means that the least common denominator is
Notice that we got the same answer as with the first method! Let's now subtract the rational expressions. Let's start by factoring the first denominator.
Therefore, the first denominator is a multiple of the second denominator. This means that the least common denominator is Thus, we need to find an equivalent fraction to whose denominator is
Multiply parentheses
Note that, once again, we obtained the same result as before. The advantage of using this method is that we do not need to factor the final expression.