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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 $\frac{x+2}{x+1}$ 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.
$\begin{array}{rl}& \phantom{=}\frac{4}{5}-\frac{1}{5}\\ \\ & =\frac{4-1}{5}\\ \\ & =\frac{3}{5}\end{array}$

### Variable expressions

The process is the same with rational expressions:
$\begin{array}{rl}& \phantom{=}\frac{7a+3}{a+2}+\frac{2a-1}{a+2}\\ \\ & =\frac{\left(7a+3\right)+\left(2a-1\right)}{a+2}\\ \\ & =\frac{7a+3+2a-1}{a+2}\\ \\ & =\frac{9a+2}{a+2}\end{array}$
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:
$\begin{array}{rl}& \phantom{=}\frac{b+1}{{b}^{2}}-\frac{4-b}{{b}^{2}}\\ \\ & =\frac{\left(b+1\right)-\left(4-b\right)}{{b}^{2}}\\ \\ & =\frac{b+1-4+b}{{b}^{2}}\\ \\ & =\frac{2b-3}{{b}^{2}}\end{array}$

### Check your understanding

Problem 1
$\frac{x+5}{x-1}+\frac{2x-3}{x-1}=$

Problem 2
Subtract.
$\frac{x+1}{2x}-\frac{5x-2}{2x}=$

## 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 $\frac{2}{3}+\frac{1}{2}$.
$\begin{array}{rl}& \phantom{=}\frac{2}{3}+\frac{1}{2}\\ \\ & =\frac{2}{3}\left(\frac{2}{2}\right)+\frac{1}{2}\left(\frac{3}{3}\right)\\ \\ & =\frac{4}{6}+\frac{3}{6}\\ \\ & =\frac{7}{6}\end{array}$
Notice that a common denominator of $6$ was needed to add the two fractions:
• The denominator in the first fraction ($3$) needed a factor of $2$.
• The denominator in the second fraction ($2$) needed a factor of $3$.
Each fraction was multiplied by a form of $1$ to obtain this.

### Variable expressions

Now let's apply this to the following example:
$\frac{1}{x-3}+\frac{2}{x+5}$
In order for the two denominators to be the same, the first needs a factor of $x+5$ and the second needs a factor of $x-3$. Let's manipulate the fractions in order to achieve this. Then, we can add as usual.
$\begin{array}{rl}& \phantom{=}\frac{1}{x-3}+\frac{2}{x+5}\\ \\ & =\frac{1}{x-3}\left(\frac{x+5}{x+5}\right)+\frac{2}{x+5}\left(\frac{x-3}{x-3}\right)\\ \\ & =\frac{1\left(x+5\right)}{\left(x-3\right)\left(x+5\right)}+\frac{2\left(x-3\right)}{\left(x+5\right)\left(x-3\right)}\\ \\ & =\frac{1\left(x+5\right)+2\left(x-3\right)}{\left(x-3\right)\left(x+5\right)}\\ \\ & =\frac{1x+5+2x-6}{\left(x-3\right)\left(x+5\right)}\\ \\ & =\frac{3x-1}{\left(x-3\right)\left(x+5\right)}\end{array}$
Notice that the first step is possible because $\frac{x+5}{x+5}$ and $\frac{x-3}{x-3}$ 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 $\left(x-3\right)\left(x+5\right)$ in the denominator, it is common to leave this in factored form.

### Check your understanding

Problem 3
$\frac{3}{x+4}+\frac{2}{x-2}=$

Problem 4
Subtract.
$\frac{2}{x-1}-\frac{5}{x}=$

### 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.
• How do i factor?
• 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.
• 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)
• 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.
• 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.