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# Algebraic Fractions: Addition and Subtraction

### Algebraic Fractions: Addition and Subtraction

Adding and subtracting algebraic fractions together allows us to express them as a single fraction.

1

Adding and subtracting algebraic fractions is much like adding and subtracting numerical fractions.

The objective is to be able to write them as a single fraction.

2

Adding fractions is simple when the denominators are the same, we just add the numerators. What is $\dfrac{5}{x+1}+\dfrac{4}{x+1}$?

3

Subtraction uses the same principle, what is $\dfrac{3}{n}-\dfrac{1+n}{n}$?

1

What happens when the denominators are not the same?

$\dfrac{x}{5} + \dfrac{x}{6}$

2

Just like with numerical fractions, the first step is to find a common denominator. What does this mean?

3

Find the lowest common denominator for $\dfrac{x}{5} + \dfrac{x}{6}$.

4

We can't just change the denominator as this would change the value of the fraction.

We need to multiply the numerator in the same way as the denominator.

5

Our first fraction is $\dfrac{x}{5}$, and we have multiplied top and bottom by 6

$\dfrac{x\times 6}{5\times 6}=\dfrac{6x}{30}$

6

What does the second fraction $\dfrac{x}{6}$ become to give it a denominator of $30$?

7

We have converted our fractions with different denominators to fractions with common denominators and can add them together as before. What is $\dfrac{x}{5} + \dfrac{x}{6} = \dfrac{6x}{30} + \dfrac{5x}{30}$?

8

Nice! The final answer is $\dfrac{11x}{30}$

This can't be simplified any further.

What is $\dfrac{x}{4} + \dfrac{x}{7}$?

Find the sum $\dfrac{2}{3b} + \dfrac{1}{2b}$

1

We can apply the same principles when the denominator contains a letter.

Let's work through this one $\dfrac{2}{3b} + \dfrac{1}{2b}$

2

What is the common denominator? $\dfrac{2}{3b} + \dfrac{1}{2b}$

3

We then multiply both fractions to make $6b$ the denominator. For the first fraction this is $\dfrac{2\times 2}{3b\times 2}=\dfrac{4}{6b}$

4

Now we can add our fractions. $\dfrac{4}{6b} + \dfrac{3}{6b}$

5

Well done - you've solved a fraction with an algebraic denominator!

The final answer is $\dfrac{7}{6b}$

What is $\dfrac{4}{3x} + \dfrac{2}{5x}$?

$\dfrac{6}{x+1}+\dfrac{x}{x+4}$

1

Let's try something a little harder. We need to take it one step at a time.

Let's try $\dfrac{6}{x+1}+\dfrac{x}{x+4}$.

2

We use the same principle.

We just need to be careful with the denominators as they each have two terms.

3

First we need to find the common denominator. This is found by multiplying the denominators together, make sure to use brackets!

4

Our common denominator is $(x+1)(x+4)$. We must multiply the numerator and denominator of the first fraction by $(x+4)$.

This makes the first fraction $\dfrac{6(x+4)}{(x+1)(x+4)}$.

5

We have converted our first fraction to $\dfrac{6(x+4)}{(x+1)(x+4)}$. What does the second fraction become?

6

We can now add the fractions together. $\dfrac{6(x+4)}{(x+1)(x+4)}+\dfrac{x(x+1)}{(x+1)(x+4)}$

7

We now have a very busy fraction $\dfrac{6(x+4)+x(x+1)}{(x+1)(x+4)}$!

We can multiply out the brackets in the numerator to simplify it a little.

8

Focusing on the numerator, multiply out and simplify $6(x+4)+x(x+1)$.

9

We can leave this as our final answer. Well done!

$\dfrac{6}{x+1}\dfrac{x}{x+4}=\dfrac{x^2+7x+24}{(x+1)(x+4)}$

1

Summary! Add algebraic fractions in the same way as numerical fractions.

Find the common denominator first. For $\dfrac{5x}{6}+\dfrac{5}{3x}$ the common denominator is $6x$.

2

Cross multiply the fractions to express them in terms of the common denominator.

$\dfrac{5x}{6}+\dfrac{5}{3x}$ $=\dfrac{5x\times x}{6x}+\dfrac{5\times 2}{6x}$ $=\dfrac{5x^2}{6x}+\dfrac{10}{6x}$

3

With a common denominator we just add the numerator.

$\dfrac{5x^2}{6x}+\dfrac{10}{6x}$ $=\dfrac{5x^2+10}{6x}$

Summary of a more complex addition: $\dfrac{x}{x+2}+\dfrac{8}{x-4}$

1

First cross multiply to get the common denominator.

$\dfrac{x(x-4)}{(x+2)(x-4)}$ $+$ $\dfrac{8(x+2)}{(x+2)(x-4)}$

2

$\dfrac{x(x-4)+8(x+2)}{(x+2)(x-4)}$

3

Multiply out the numerator

$\dfrac{x^2-4x+8x+16)}{(x+2)(x-4)}$

4

Gather the like terms together for the final answer

$\dfrac{x^2+4x+16}{(x+2)(x-4)}$