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# Adding and Subtracting Rational Expressions

Rational expression is that where the numerator and the denominator or both of them are polynomials. Keep in mind any value of the variable that makes a denominator in a rational equation equal to zero cannot be a solution of the equation. We can directly add or subtract the numerators if the denominators are the same.

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## Adding and Subtracting Rational Expressions with Like Denominators

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We will be adding and subtracting rational expressions, by keeping a like denominators same and performing the operation on numerator. Then, simplify the terms and factor the numerator, and write the expressions in a lowest terms.

Rational expressions with a like denominators are added and subtracted as:

$\frac{a}{b}$ + $\frac{c}{b}$ = $\frac{a + c}{b}$

and $\frac{a}{b}$ - $\frac{c}{b}$ = $\frac{a - c}{b}$

Below you could see examples for adding and subtracting rational expressions with like denominators.

### Solved Examples

Question 1: Simplify $\frac{9x + 2}{x + 2}$ + $\frac{4x - 7}{x + 2}$
Solution:
Given $\frac{9x + 2}{x + 2}$ + $\frac{4x - 7}{x + 2}$

= $\frac{9x + 2 + (4x - 7)}{x + 2}$

= $\frac{9x + 2 + 4x - 7}{x + 2}$

= $\frac{13x - 5}{x + 2}$

Question 2: Simplify $\frac{13x + 5}{x - 4}$ - $\frac{2x - 8}{x - 4}$
Solution:
Given $\frac{13x + 5}{x - 4}$ - $\frac{2x - 8}{x - 4}$

= $\frac{13x + 5 - (2x - 8)}{x - 4}$

= $\frac{13x + 5 - 2x + 8}{x - 4}$

= $\frac{11x + 13}{x - 4}$

## Adding and Subtracting Rational Expressions with Unlike Denominators

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To add and subtract two rational expressions with unlike denominators, first we rewrite the expressions with a common denominator because two rational expression is added or subtracted only if they have same denominators. Find the LCD(Least Common Denominator) of two rational expressions, then factor their denominator.

Below you could see some examples on adding and subtracting rational expressions with unlike denominators.

### Solved Examples

Question 1: $\frac{3x + 2}{5x - 2}$ + $\frac{8x - 3}{5x + 2}$
Solution:

Take L.C.M of (5x - 2) and (5x + 2) is (5x)2 – 22.

So, given expression is reduced as

$\frac{(3x+2)(5x+2)}{(5x)^2 – 2^2}$ + $\frac{(8x - 3)(5x-2)}{(5x)^2 – 2^2}$

= $\frac{(15x^2 + 16x + 4) + (40x^2 - 31x + 6)}{5x^2 - 2^2}$

= $\frac{55x^2 - 15x + 10}{5x^2 - 2^2}$

= $\frac{55x^2 - 15x + 10}{25x^2 - 4}$

Question 2: Solve the polynomial expression $\frac{2}{x - 1}$ - $\frac{1}{x + 1}$
Solution:

Given rational expression, $\frac{2}{x - 1}$ - $\frac{1}{x + 1}$

In this expression, the denominators are not common. To make common denominator, take the LCM.

LCM ((x - 1) and (x + 1)) = (x - 1)(x + 1) = x2 - 1

Now, we get

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

$\frac{2x + 2 - x + 1}{x^2 - 1}$
$\frac{x + 3}{x^2 - 1}$

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