ERF - Adding and Subtracting Rational Expressions Lesson

Adding and Subtracting Rational Expressions

We can add and subtract rational expressions the same way we add and subtract numerical fractions.  We must have common denominators first. Then, simply add or subtract the numerators and keep the denominator.

Given:  $\frac{4}{15}-\frac{7}{15}$
$\frac{4-7}{15}$	Subtract the numerators and write as one fraction.
$-\frac{3}{15}=-\frac{1}{5}$	Check to see if the fraction can be reduced further.  This fraction is reducible by 3.

Adding and subtracting rational expressions follows the same process.  The first step in adding and subtracting rational expressions is to make sure you have common denominators. 

If the denominators are already the same, simply add or subtract the numerator and keep your denominator. When subtracting, remember to distribute the negative sign to all terms in the second numerator first, then combine like terms.

Adding rational expressions:	Subtracting rational expressions:
$\frac{x-3}{x+1}+\frac{2x-4}{x+1}=\frac{3x-7}{x+1}$	$\frac{3x+7}{2x}-\frac{x-4}{2x}=\frac{3x+7-x+4}{2x}=\frac{2x+11}{2x}$

Here is a video to demonstrate this concept.

 

What happens when we are trying and/or subtract rational expressions without "common" or "like" denominators? Consider the problem $\frac{3}{10}+\frac{1}{6}$. First, find the least common denominator (LCD).  Remember in finding the LCD, you are really finding the "least common multiple" (LCM) of both denominators. In this problem, the LCD equals 30. Once we find the LCD, then we need to change each individual fraction to be equivalent to respective fractions that have a denominator equal to 30. Once you have common denominators in all fractions, then you add the numerators together and simplify if possible.

Given:  $\frac{3}{10}+\frac{1}{6}$	The least common denominator (LCD) is 30.
$\frac{3(3)}{10(3)}+\frac{1(5)}{6(5)}=\frac{9}{30}+\frac{5}{30}$	Multiply the numerator and denominator of each fraction by the same factor to make the denominator equal to the LCD.
$\frac{14}{30}=\frac{7}{15}$	Add the numerators and write the answer as one fraction.  Check to see if the fraction can be reduced further. This fraction is reducible by 2.

 

When thinking about denominators like (x + 2) or (x - 3) it becomes important to understand what makes a LCD. In the problem above, we found the LCD was equal to 30 because it is the smallest number that both 10 and 6 divide into, but how do we create that number if it isn't obvious? (Hint: Think about prime factorization.)

When dealing with rational expressions, factoring is key. We must find all of the factors of each denominator to know what the LCD should be.

Let's try one example.

Find the LCD for $\frac{3}{5a}\:\&\:\frac{b}{4a^2}$.

$5a=5\cdot a \\ 4a^2=4\cdot a\cdot a$

The only common factor is a, so cancel them and multiply all the remaining factors: $a\cdot5\cdot4\cdot a=20a^2$.

Therefore, the LCD is $20a^2$.

To add the fractions together we create common denominators for both expressions by multiply the numerator and denominator of the first expression by (4a) and the second expression by (5).  Then, add the numerators by combining like terms. 

$\frac{3}{5a^{}}+\frac{b}{4a^2}=\frac{3(4a)}{5a(4a)}+\frac{(5)b}{(5)4a^2}=\frac{12a+5b}{20a^2}$ .

Watch the following 3 videos to review and apply the concept of finding the LCD of rational expressions. Then complete the practice problems below.

Lowest Common Denominator Practice

Example: Add rational expressions.

$\frac{3a}{4a-20}+\frac{9a}{6a-30}$	Given: Add rational expression
$\frac{3a}{4(a-5)}+\frac{9a}{6(a-5)}$	Factor denominators to find LCD.
$(\frac{3}{3})\frac{3a}{4(a-5)}+(\frac{2}{2})\frac{9a}{6(a-5)}$	LCD is 12(a - 5). Multiply first fraction by 3 and second fraction by 2.
$\frac{9a}{12(a-5)}+\frac{18a}{12(a-5)}$	Add numerators
$\frac{27a}{12(a-5)}=\frac{9a}{4(a-5)},x\ne5$	Reduce by 3, state excluded values

Example: Subtract rational expressions.

$\frac{2x}{x^2-1}-\frac{3x-1}{x^2+5x+4}$	Given: Subtract the rational expressions.
$\frac{2x}{x^2-1}+\frac{-3x+1}{x^2+5x+4}$	Distribute negative to second numerator
$\frac{2x}{(x+1)(x-1)}+\frac{-3x+1}{(x+4)(x+1)}$	Factor denominators to find LCD. LCD is (x + 1)(x - 1)(x + 4)
$(\frac{x+4}{x+4})\frac{2x}{(x+1)(x-1)}+\frac{-3x+1}{(x+4)(x+1)}(\frac{x-1}{x-1})$	Multiply first fraction by (x + 4) and second fraction by (x - 1)
$\frac{2x^2+8x}{(x+4)(x+1)(x-1)}+\frac{-3x^2+4x-1}{(x+4)(x+1)(x-1)}$	Add numerators by combining like terms.
$\frac{-x^2+12x-1}{(x+4)(x+1)(x-1)},x\ne-4,-1,1$	Final solution, state excluded values.

Now watch and take notes on the following 5 videos that discuss and model the concepts involved in adding and subtracting rational expressions with monomials, binomials, multiple variables, and other unknown expressions. After watching these teaching videos, practice items are provided for you to complete to master these concepts.

Adding Rational Expressions Practice

Subtracting Rational Expressions Practice

Complex Fractions

There are times that you come across problems that involve "nested fractions" (fractions within fractions). In trying to simplify these types of "complex fractions," often times you will need to use all of the operational properties of rational expressions that you have learned to this point.

Watch the following video describing the process of simplifying complex fractions, and after the video there are some practice problems provided for you to work to attain mastery.

Complex Fractions and Nested Rational Expressions Practice

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