>Fractions and Factorization>How to Add and Subtract Fractions with x in the Denominator

How to Add and Subtract Fractions with x in the Denominator

Similar to other fractions, you need a common denominator to add or subtract fractions with variables. That means you need to rewrite the fractions to get a common denominator.

Rule

Add and Subtract Fractions when $x$ Is in the Denominator

Take a look at the examples below while you read through the steps. In some of the examples, you don’t need to factorize, so you can start at Item 2.

1.: If one or more of the denominators can be factorized, you should factorize them.
2.: The common denominator is the product of the denominators.
3.: Multiply the numerator and denominator of each fraction with the factors from the common denominator that aren’t in their own denominator.
4.: Expand the numerators and denominators.
5.: Now you can finally perform the addition or subtraction. After that, factorize and simplify the resulting fraction.

Note! When the fractions have multiple terms in the numerators, it helps to imagine parentheses around them. Then you can use the rules for parentheses in your calculations.

Example 1

\begin{array}{l} \frac{2}{2 x} + \frac{3}{x^{2}} & = \frac{2 \cdot x}{2 x \cdot x} + \frac{3 \cdot 2}{x^{2} \cdot 2} \\ = \frac{2 x}{2 x^{2}} + \frac{6}{2 x^{2}} \\ = \frac{2 x + 6}{2 x^{2}} \end{array}

Example 2

\begin{array}{l} \frac{5}{(x + 1)} - \frac{1}{x} & = \frac{5 \cdot x}{(x + 1) \cdot x} - \frac{x + 1}{x \cdot (x + 1)} \\ = \frac{5 x - (x + 1)}{x (x + 1)} \\ = \frac{4 x - 1}{x (x + 1)} \end{array}

Example 3

\begin{array}{l} \frac{2 x + 2}{{(x + 1)}^{2}} - \frac{4}{x^{2} - 1} \\ = \frac{2 (x + 1)}{(x + 1) (x + 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2 (x+1)}{(x+1) (x + 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2}{(x + 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2 (x - 1)}{(x + 1) (x - 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2 x - 2}{(x - 1) (x + 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2 x - 2 - 4}{(x - 1) (x + 1)} \\ = \frac{2 x - 6}{(x - 1) (x + 1)} \\ = \frac{2 (x - 3)}{(x + 1) (x - 1)} \end{array}

\begin{array}{l} \frac{2 x + 2}{{(x + 1)}^{2}} - \frac{4}{x^{2} - 1} & = \frac{2 (x + 1)}{(x + 1) (x + 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2 (x+1)}{(x+1) (x + 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2}{(x + 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2 (x - 1)}{(x + 1) (x - 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2 x - 2}{(x - 1) (x + 1)} - \frac{4}{(x - 1) (x + 1)} \\ = \frac{2 x - 2 - 4}{(x - 1) (x + 1)} \\ = \frac{2 x - 6}{(x - 1) (x + 1)} \\ = \frac{2 (x - 3)}{(x + 1) (x - 1)} \end{array}

Want to know more?Sign UpIt's free!

How to Divide Fractions by Canceling

How to Factor Variable Expressions

Go back

Add and Subtract Fractions when x Is in the Denominator

Add and Subtract Fractions when $x$ Is in the Denominator