The process for adding and subtracting algebraic fractions depends on whether the fractions are like (with the same denominator) or unlike (with different denominators). If the fractions are like fractions, we have to start by calculating the lowest common denominator.

Here, we will look at the steps that we can follow to add and subtract both like fractions and unlike fractions. Then we will solve some practical examples.

## Steps to add and subtract like algebraic fractions

Like fractions are fractions that have the same denominators. For example, the fractions $latex \frac{1}{x+1}$ and $latex \frac{3}{x+1}$ are like fractions.

To add and subtract these types of fractions, we simply have to “combine” the fractions by using a single denominator and simplifying the numerator operations.

Therefore, we can follow the steps below to add and subtract like algebraic fractions:

#### 1. Write the fractions using a single denominator.

Since the denominator is the same, we can “combine” the fractions into one.

#### 2. Perform the operations in the numerator.

Simplify and combine like terms.

#### 3. Simplify the end result.

## Steps to add and subtract unlike algebraic fractions

Unlike fractions are fractions that have different denominators. For example, the fractions $latex \frac{1}{x+2}$ and $latex \frac{3}{x-3}$ are unlike fractions.

To add and subtract these types of fractions, we have to start by finding the least common multiple of the denominators. Then, we rewrite the fractions to get like fractions.

Therefore, we can add and subtract unlike algebraic fractions by applying the following steps:

#### 1. Find the lowest common denominator (LCD).

#### 2. Use the lowest common denominator to write like fractions.

The new denominator of each fraction is the LCD. The new numerator of each fraction is equal to the LCD divided by the denominator and multiplied by the numerator of the original fraction.

#### 3. Write the fractions using a single denominator.

Since the denominators of the fractions from step 2 are the same, we combine them to get a single fraction.

#### 4. Perform the operations of the numerator.

Simplify and combine like terms.

#### 5. Simplify the final fraction.

## Examples of adding and subtracting algebraic fractions

**EXAMPLE **1

**EXAMPLE**

What is the result of the addition of fractions? $$\frac{3x+2}{5}+\frac{x}{5}$$

##### Solution

We have an addition of like fractions. Therefore, we can write it using a single denominator as follows:

$$\frac{3x+2}{5}+\frac{x}{5}=\frac{3x+2+x}{5}$$

Now, we combine like terms in the numerator:

$$\frac{3x+2+x}{5}=\frac{4x+2}{5}$$

The fraction can no longer be simplified, so $latex \frac{4x+2}{5}$ is our final result.

**EXAMPLE **2

**EXAMPLE**Find the result of subtracting the fractions: $$\frac{4x+1}{x+2}-\frac{3-2x}{x+2}$$

##### Solution

In this case, we have a subtraction of like fractions, so we can combine them to form a single fraction:

$$\frac{4x+1}{x+2}-\frac{3-2x}{x+2}=\frac{4x+1-(3-2x)}{x+2}$$

$$=\frac{4x+1-3+2x}{x+2}$$

Combining like terms in the numerator, we have:

$$\frac{4x+1-3+2x}{x+2}=\frac{6x-2}{x+2}$$

We can simplify the numerator: $latex \frac{2(3x-1)}{x+2}$ and this is our final result.

**EXAMPLE **3

**EXAMPLE**Find the result of: $$\frac{3}{x-1}+\frac{2}{x+3}$$

##### Solution

In this case, we have an addition of unlike fractions. Therefore, we have to start by finding the lowest common denominator.

The lowest common denominator is $latex (x-1)(x+3)$ since it can be divided by both denominators without leaving a remainder.

Then, we form like fractions by dividing the LCD by each denominator and multiplying by the numerator. Thus, we have:

$$\frac{3}{x-1}+\frac{2}{x+3}=\frac{3(x+3)}{(x-1)(x+3)}+\frac{2(x-1)}{(x-1)(x+3)}$$

Combining the like fractions and performing the numerator operations, we have:

$$=\frac{3(x+3)+2(x-1)}{(x-1)(x+3)}$$

$$=\frac{3x+9+2x-2}{(x-1)(x+3)}$$

$$=\frac{5x+7}{(x-1)(x+3)}$$

We can no longer simplify the fraction. Then, $latex \frac{5x+7}{(x-1)(x+3)}$ is our final result.

**EXAMPLE **4

**EXAMPLE**Solve the following subtraction: $$\frac{4}{x+6}-\frac{2}{x+7}$$

##### Solution

Since we have a subtraction of unlike fractions, we start by finding the lowest common denominator.

In this case, the lowest common denominator is $latex (x+6)(x+7)$. Therefore, we form like fractions using that denominator:

$$\frac{4}{x+6}-\frac{2}{x+7}=\frac{4(x+7)}{(x+6)(x+7)}-\frac{2(x+6)}{(x+6)(x+7)}$$

Now, we can form a single fraction and simplify the numerator:

$$=\frac{4(x+7)-2(x+6)}{(x+6)(x+7)}$$

$$=\frac{4x+28-2x-12}{(x+6)(x+7)}$$

$$=\frac{2x+16}{(x+6)(x+7)}$$

$$=\frac{2(x+8)}{(x+6)(x+7)}$$

**EXAMPLE **5

**EXAMPLE**Find the result of the following addition of fractions: $$\frac{2x}{x^2+3x+2}+\frac{3}{x+1}$$

##### Solution

To find the lowest common denominator, we can start by recognizing that:

$$x^2+3x+2=(x+1)(x+2)$$

This means that $latex (x+1)$ is a factor of $latex x^2+3x+2$ and $latex (x+1)(x+2)$ is the lowest common denominator. Therefore, we have:

$$\frac{2x}{x^2+3x+2}+\frac{3}{x+1}=\frac{2x}{(x+1)(x+2)}+\frac{3(x+2)}{(x+1)(x+2)}$$

Combining the like fractions and performing the numerator operations, we have:

$$=\frac{2x+3(x+2)}{(x+1)(x+2)}$$

$$=\frac{2x+3x+6}{(x+1)(x+2)}$$

$$=\frac{5x+6}{(x+1)(x+2)}$$

**EXAMPLE **6

**EXAMPLE**Find the result of the following addition and subtraction of fractions: $$x+7+\frac{1}{x-4}-\frac{5}{x+1}$$

##### Solution

We can write this expression as follows:

$$\frac{x+7}{1}+\frac{1}{x-4}-\frac{5}{x+1}$$

Therefore, the lowest common denominator is $latex (x-4)(x+1)$ and we have the following:

Then, we form like fractions by dividing the LCD by each denominator and multiplying by the numerator. Thus, we have:

$$\frac{x+7}{1}+\frac{1}{x-4}-\frac{5}{x+1}$$

$$=\frac{(x+7)(x-4)(x+1)+(x+1)-5(x-4)}{(x-4)(x+1)}$$

$$=\frac{x^3+4x^2-25x-28+x+1-5x+20}{(x-4)(x+1)}$$

$$=\frac{x^3+4x^2-29x-7}{(x-4)(x+1)}$$

You can explore additional exercises on this topic in these articles: Adding Algebraic Fractions – Examples and Practice, Subtracting Algebraic Fractions – Examples and Practice.

## Addition and subtraction of algebraic fractions – Practice problems

#### What is the numerator of the fraction obtained by performing the following operation? $$\frac{5x}{2x^2+3x-5}-\frac{1}{x-1}-\frac{2}{2x+5}$$

Write the numerator in the input box.

## See also

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