I still dont get it, is there a way you could break it down more

If I understand properly, you're asking why not to distribute the expressions. For example, (x-2) to (x+4). If that is so, it isn't necessary to as the article states it is common to leave it in factored form.

I'm still stuck because I have two numerators with differing variables. I have to subtract 2x and 3y. Do I just keep it as 2x-3y?

Yes, 2x - 3y is as simplified as you can go. They are unlike terms, so you can't actually subtract.

So why don't we simplify even more when you reach 9a+2/a+2 as the answer. I mean it says to simplify? I don't get it.

There is no common FACTOR (something being multiplied) in both numerator and denominator. Instead, they each consist of 2 TERMS (things being added or subtracted) which must therefore be used as a single quantity, for example, (9a + 2) or ( a + 2 ). If the problem had been 9a*2 / a*2, it could have been simplified by dividing out a*2, to get 9.

Hi 3081724, To factor monomials: https://www.khanacademy.org/math/algebra/polynomial-factorization/factoring-monomials/v/ways-to-factor-a-monomial To factor polynomials: http://www.dummies.com/education/math/pre-calculus/how-to-factor-a-polynomial-expression/ To factor trinomials: http://www.mathwarehouse.com/algebra/factor/how-to-factor-trinomials-step-by-step.php Hope that helps! - JK

Does the denominator have to be the same to add and subtract, or can they be different?

With all types of fractions, you must have a common denominator to add them.

what would you do if you were going to do r/r-1 - r-1/r

common denominator is r(r-1) so multiply numerator of first times r and second times r-1 and add on the common denominator (r*r-(r-1)^2)/(r(r-1)), distribute top and simplify r*2-(r^2-2r+1)=2r-1, so answer is (2r-1)/(r(r-1))

what would you do if there were different denominators?

look down through the steps a little bit more carefully

In the check your understanding questions, if you simplify the denominator, you get marked as wrong. Is there a reason for this?

When you simplify the denominator, did you also simplify the numerator accordingly? Did you simplify both of them all the way down? Could you give 1 example that marked wrong.

Main content

Algebra (all content)

Course: Algebra (all content) > Unit 13

Lesson 4: Adding & subtracting rational expressions

Intro to adding & subtracting rational expressions

CCSS.Math:

HSA.APR.D.7

HSA.APR.D

Google Classroom

Learn how to add or subtract two rational expressions into a single expression.

What you should be familiar with before taking this lesson

A rational expression is a quotient of two polynomials. For example, the expression start fraction, x, plus, 2, divided by, x, plus, 1, end fraction is a rational expression.

If you are unfamiliar with rational expressions, you may want to check out our intro to rational expressions.

What you will learn in this lesson

In this lesson, you will learn how to add and subtract rational expressions.

Adding and subtracting rational expressions (common denominators)

Numerical fractions

We can add and subtract rational expressions in much the same way as we add and subtract numerical fractions.

To add or subtract two numerical fractions with the same denominator, we simply add or subtract the numerators, and write the result over the common denominator.

\begin{aligned} &\phantom{=}\dfrac{\blueE4}{\purpleD5}-\dfrac{\blueE1}{\purpleD5} \\\\ &=\dfrac{\blueE{4}-\blueE{1}}{\purpleD 5} \\\\ &=\dfrac{3}{5} \end{aligned}

Variable expressions

The process is the same with rational expressions:

\begin{aligned} &\phantom{=}\dfrac{\blueE{7a+3}}{\purpleD{a+2}}+\dfrac{\blueE{2a-1}}{\purpleD{a+2}} \\\\ &=\dfrac{(\blueE{7a+3})+(\blueE{2a-1})}{\purpleD{a+2}} \\\\ &=\dfrac{{7a+3}+{2a-1}}{{a+2}} \\\\ &=\dfrac{9a+2}{a+2} \end{aligned}

It is good practice to place the numerators in parentheses, especially when subtracting rational expressions. This way, we are reminded to distribute the negative sign!

For example:

\begin{aligned} &\phantom{=}\dfrac{\blueE{b+1}}{\purpleD{b^2}}-\dfrac{\blueE{4-b}}{\purpleD{b^2}} \\\\ &=\dfrac{(\blueE{b+1})-(\blueE{4-b})}{\purpleD{b^2}} \\\\ &=\dfrac{b+1-4+b}{{b^2}} \\\\ &=\dfrac{2b-3}{b^2} \end{aligned}

Check your understanding

Problem 1

Add.

start fraction, x, plus, 5, divided by, x, minus, 1, end fraction, plus, start fraction, 2, x, minus, 3, divided by, x, minus, 1, end fraction, equals

Problem 2

Subtract.

start fraction, x, plus, 1, divided by, 2, x, end fraction, minus, start fraction, 5, x, minus, 2, divided by, 2, x, end fraction, equals

Adding and subtracting rational expressions (different denominators)

Numerical fractions

To understand how to add or subtract rational expressions with different denominators, let's first examine how this is done with numerical fractions.

For example, let's find start fraction, 2, divided by, 3, end fraction, plus, start fraction, 1, divided by, 2, end fraction.

\begin{aligned} &\phantom{=}\dfrac{2}{\blueE3}+\dfrac{1}{\tealE2} \\\\ &=\dfrac{2}{\blueE3} \left(\tealE{\dfrac{2}{2}}\right)+\dfrac{1}{\tealE2}\left( \blueE{\dfrac{3}{3}}\right) \\\\ &=\dfrac{4}{6}+\dfrac{3}{6} \\\\ &=\dfrac{7}{6} \end{aligned}

Notice that a common denominator of 6 was needed to add the two fractions:

The denominator in the first fraction (start color #0c7f99, 3, end color #0c7f99) needed a factor of start color #208170, 2, end color #208170.
The denominator in the second fraction (start color #208170, 2, end color #208170) needed a factor of start color #0c7f99, 3, end color #0c7f99.

Each fraction was multiplied by a form of 1 to obtain this.

Variable expressions

Now let's apply this to the following example:

start fraction, 1, divided by, start color #0c7f99, x, minus, 3, end color #0c7f99, end fraction, plus, start fraction, 2, divided by, start color #208170, x, plus, 5, end color #208170, end fraction

In order for the two denominators to be the same, the first needs a factor of start color #208170, x, plus, 5, end color #208170 and the second needs a factor of start color #0c7f99, x, minus, 3, end color #0c7f99. Let's manipulate the fractions in order to achieve this. Then, we can add as usual.

\begin{aligned} &\phantom{=}{\dfrac{1}{\blueE{x-3}}+\dfrac{2}{\tealE{x+5}}} \\\\ &=\dfrac{1}{\blueE{x-3}}{\left(\tealE{\dfrac{x+5}{x+5}}\right)}+\dfrac{2}{\tealE{x+5}}{\left(\blueE{\dfrac{x-3}{x-3}}\right)} \\\\ &=\dfrac{1(x+5)}{(x-3)(x+5)}+\dfrac{2(x-3)}{(x+5)(x-3)} \\\\ &=\dfrac{1(x+5)+2(x-3)}{(x-3)(x+5)} \\\\ &=\dfrac{1x+5+2x-6}{(x-3)(x+5)} \\\\ &=\dfrac{3x-1}{(x-3)(x+5)} \end{aligned}

Notice that the first step is possible because start fraction, x, plus, 5, divided by, x, plus, 5, end fraction and start fraction, x, minus, 3, divided by, x, minus, 3, end fraction are equal to 1, and multiplication by 1 does not change the value of the expression!

[What about the restricted values?]

In the last two steps, we rewrote the numerator. While you can also expand left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 5, right parenthesis in the denominator, it is common to leave this in factored form.

Check your understanding

Problem 3

Add.

start fraction, 3, divided by, x, plus, 4, end fraction, plus, start fraction, 2, divided by, x, minus, 2, end fraction, equals

Problem 4

Subtract.

start fraction, 2, divided by, x, minus, 1, end fraction, minus, start fraction, 5, divided by, x, end fraction, equals

What's next?

Our next article covers more challenging examples of adding and subtracting rational expressions.

You will learn about the least common denominator, and why it is important to use this as the common denominator when adding or subtracting rational expressions.

Want to join the conversation?

Sort by:

bjames
Posted 6 years ago. Direct link to bjames's post “I still dont get it, is t...”
I still dont get it, is there a way you could break it down more
Button navigates to signup pageButton navigates to signup page
(11 votes)
Answer
- AK
  Posted 5 years ago. Direct link to AK's post “If I understand properly,...”
  If I understand properly, you're asking why not to distribute the expressions. For example, (x-2) to (x+4). If that is so, it isn't necessary to as the article states it is common to leave it in factored form.
  Comment on AK's post “If I understand properly,...”
  (5 votes)
Madison Fenner
Posted 6 years ago. Direct link to Madison Fenner's post “I'm still stuck because I...”
I'm still stuck because I have two numerators with differing variables. I have to subtract 2x and 3y. Do I just keep it as 2x-3y?
Button navigates to signup pageComment on Madison Fenner's post “I'm still stuck because I...”
(5 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “Yes, 2x - 3y is as simpli...”
  Yes, 2x - 3y is as simplified as you can go. They are unlike terms, so you can't actually subtract.
  Comment on Kim Seidel's post “Yes, 2x - 3y is as simpli...”
  (6 votes)
sadeghmoghaddamnarge
Posted 6 years ago. Direct link to sadeghmoghaddamnarge's post “So why don't we simplify ...”
So why don't we simplify even more when you reach 9a+2/a+2 as the answer. I mean it says to
simplify? I don't get it.
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Judith Gibson
  Posted 6 years ago. Direct link to Judith Gibson's post “There is no common FACTOR...”
  There is no common FACTOR (something being multiplied) in both numerator and denominator. Instead, they each consist of 2 TERMS (things being added or subtracted) which must therefore be used as a single quantity, for example, (9a + 2) or ( a + 2 ).
  If the problem had been 9a*2 / a*2, it could have been simplified by dividing out a*2, to get 9.
  Comment on Judith Gibson's post “There is no common FACTOR...”
  (4 votes)
3081724
Posted 6 years ago. Direct link to 3081724's post “How do i factor?”
How do i factor?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- [SKLZ] ▁ ▂ ▄ ▅ ▆ ▇ █ נк █ ▇ ▆ ▅ ▄ ▂ ▁
  Posted 6 years ago. Direct link to [SKLZ] ▁ ▂ ▄ ▅ ▆ ▇ █ נк █ ▇ ▆ ▅ ▄ ▂ ▁'s post “Hi 3081724, To factor mo...”
  Hi 3081724,
  
  To factor monomials:
  https://www.khanacademy.org/math/algebra/polynomial-factorization/factoring-monomials/v/ways-to-factor-a-monomial
  
  To factor polynomials:
  http://www.dummies.com/education/math/pre-calculus/how-to-factor-a-polynomial-expression/
  
  To factor trinomials:
  http://www.mathwarehouse.com/algebra/factor/how-to-factor-trinomials-step-by-step.php
  
  Hope that helps!
  - JK
  Comment on [SKLZ] ▁ ▂ ▄ ▅ ▆ ▇ █ נк █ ▇ ▆ ▅ ▄ ▂ ▁'s post “Hi 3081724, To factor mo...”
  (6 votes)
‧͙⁺˚*･༓𝐸𝓁𝒾𝓏𝒶𝒷𝑒𝓉𝒽 𝒴.༓･*˚⁺‧͙
Posted 4 years ago. Direct link to ‧͙⁺˚*･༓𝐸𝓁𝒾𝓏𝒶𝒷𝑒𝓉𝒽 𝒴.༓･*˚⁺‧͙'s post “If there are rational exp...”
If there are rational expressions, then are there irrational expressions?
Button navigates to signup pageComment on ‧͙⁺˚*･༓𝐸𝓁𝒾𝓏𝒶𝒷𝑒𝓉𝒽 𝒴.༓･*˚⁺‧͙'s post “If there are rational exp...”
(5 votes)
Answer
- Jervin Sevilla
  Posted 3 years ago. Direct link to Jervin Sevilla's post “An irrational expression ...”
  An irrational expression cannot be expressed as the quotient of 2 polynomials (e.g. 2^x, log(x)/x) but "irrational" is not usually used for of expressions or functions.
  Button navigates to signup page
  (1 vote)
Terrabronson
Posted 6 years ago. Direct link to Terrabronson's post “Does the denominator have...”
Does the denominator have to be the same to add and subtract, or can they be different?
Button navigates to signup pageComment on Terrabronson's post “Does the denominator have...”
(2 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “With all types of fractio...”
  With all types of fractions, you must have a common denominator to add them.
  Button navigates to signup page
  (5 votes)
ElfPrincess318
Posted 6 years ago. Direct link to ElfPrincess318's post “How can you find the leas...”
How can you find the least common multiple of polynomials?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “Try this video: https://...”
  Try this video: https://www.khanacademy.org/math/algebra2/rational-expressions-equations-and-functions/adding-and-subtracting-rational-expressions/v/least-common-multiples-of-polynomials
  Button navigates to signup page
  (1 vote)
spol29
Posted 2 years ago. Direct link to spol29's post “what would you do if you ...”
what would you do if you were going to do r/r-1 - r-1/r
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- David Severin
  Posted 2 years ago. Direct link to David Severin's post “common denominator is r(r...”
  common denominator is r(r-1) so multiply numerator of first times r and second times r-1 and add on the common denominator (r*r-(r-1)^2)/(r(r-1)), distribute top and simplify r*2-(r^2-2r+1)=2r-1, so answer is (2r-1)/(r(r-1))
  Button navigates to signup page
  (2 votes)
justin josephson
Posted 4 years ago. Direct link to justin josephson's post “what would you do if ther...”
what would you do if there were different denominators?
Button navigates to signup pageComment on justin josephson's post “what would you do if ther...”
(1 vote)
Answer
- Graham
  Posted 3 years ago. Direct link to Graham's post “look down through the ste...”
  look down through the steps a little bit more carefully
  Button navigates to signup page
  (2 votes)
Eddy Some
Posted 5 years ago. Direct link to Eddy Some's post “In the check your underst...”
In the check your understanding questions, if you simplify the denominator, you get marked as wrong. Is there a reason for this?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Thien D Ho
  Posted 5 years ago. Direct link to Thien D Ho's post “When you simplify the den...”
  When you simplify the denominator, did you also simplify the numerator accordingly? Did you simplify both of them all the way down? Could you give 1 example that marked wrong.
  Button navigates to signup page
  (2 votes)