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# Intro to adding & subtracting rational expressions

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}

Problem 1
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!
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.

Problem 3
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?

• 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.
• If there are rational expressions, then are there irrational expressions?
• 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.
(1 vote)
• How do i factor?
• 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.
• How can you find the least common multiple of polynomials?
• what would you do if you were going to do r/r-1 - r-1/r
(1 vote)
• 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?
(1 vote)
• 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?
(1 vote)
• 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.