# Add and subtract rational expressions with and without common denominators

Adding and subtracting rational expressions is identical to adding and subtracting with integers. Recall that when adding with a common denominator we add the numerators and keep the denominator. This is the same process used with rational expressions. Remember to reduce, if possible, your final answer.

Example 1:

Determine :

$\frac{x-4}{x^2-2x-8}+\frac{x+8}{x^2-2x-8}$

Solution :

Same denominator, add numerators, combine like terms

$\frac{2x+4}{x^2-2x-8}$

Factor numerator and denominator

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

Divide out (x +2), so the solution is

$\frac{2}{x-4}$

Subtraction with common denominator follows the same pattern, though the subtraction can cause problems if we are not careful with it. To avoid sign errors we will first distribute the subtraction through the numerator. Then we can treat it like an addition problem. This process is the same as “add the opposite” we saw when subtracting with negatives.

Example 2:

Determine:

$\frac{6x-12}{3x-6}-\frac{15x-6}{3x-6}$

Solution:

Add the opposite of the second fraction (distribute negative)

$\frac{6x-12}{3x-6}+\frac{-15x+6)}{3x-6}$

$\frac{-9x-6}{3x-6}$

Factor numerator and denominator

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

Divide out common factor of 3, and the solution is

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

Also Read : Multiply and divide expressions using scientific notation and exponent properties

World View Note: The Rhind papyrus of Egypt from 1650 BC gives some of the earliest known symbols for addition and subtraction, a pair of legs walking in the direction one reads for addition, and a pair of legs walking in the opposite direction for subtraction.

When we don’t have a common denominator we will have to find the least common denominator (LCD) and build up each fraction so the denominators match. The following example shows this process with integers.

Example 3:

Determine:

$\frac{5}{6}+\frac{1}{4}$

Solution:

The LCD is 12. Build up, multiply 6 by 2 and 4 by 3

$(\frac{2}{2})\frac{5}{6}+\frac{1}{4}(\frac{3}{3})$

Multiply

$\frac{10}{12}+\frac{3}{12}$

Add numerators, and the solution is $\frac{13}{12}$.

The same process is used with variables.

Example 4:

Determine:

$\frac{7a}{3a^2b}+\frac{4b}{6ab^4}$

Solution:

The LCD is $6a^2b^4$. We will then build up each fraction

Multiply first fraction by $2b^3$ and second by a

$(\frac{2b^3}{2b^3})(\frac{7a}{3a^2b})+(\frac{4b}{6ab^4})(\frac{a}{a})$

$\frac{14ab^3}{6a^2b^4}+\frac{4ab}{6a^2b^4}$

Add numerators, no like terms to combine

$\frac{14ab^3+4ab}{6a^2b^4}$

Factor numerator

$\frac{2ab(7b^3+2)}{6a^2b^4}$

And our solution is

$\frac{7b^3+2}{3ab^3}$

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