# Adding Rational Expressions

Adding and subtracting rational expressions are similar to adding and subtracting numerical ratios.

In order to add or subtract a rational expression, a common denominator must be found first, and then the operation can be carried out in the numerator.

With like denominators, simply add the two numerators to find the sum.

Example 1:

Add the numerators (the denominator does not change).

With unlike denominators

First find the common denominator

By multiplying denominators

Example 2: Single Term Ratios
$\frac{3}{{2}{x}}+\frac{5}{{7}}$

$\frac{{7}}{{7}}×\frac{3}{{2}{x}}+\frac{{2}{x}}{2x}×\frac{5}{7}$     Multiply each ratio by one using the other denominator.

$\frac{21}{{14}{x}}+\frac{10x}{{14}{x}}$ Multiply across.

$\frac{10x+21}{14x}$      Add the numerators.

By finding least common multiple of the denominators

Example 3: Single Term Ratios
$\frac{7}{{8}{x}}+\frac{16}{{5}{x}}$

The least common multiple (LCM) from 8x and 5x is 40x

Multiply the ratios by one to get a common denominator.

$\frac{35}{{40}{x}}+\frac{128}{{40}{x}}$ Multiply across.

$\frac{163}{40x}$   Add the numerators, simplify if possible.

163 and 40 are relatively prime, so this ratio cannot be simplified.

Example 4: Factoring trinomials in the denominator.

$\frac{2x}{{x}^{2}+5x-24}+\frac{4}{x-3}$

x2 + 5x - 24 = (x + 8)(x - 3)Factor the denominator.

The LCM is (x + 8)(x - 3).

$\frac{2x}{\left(x+8\right)\left(x-3\right)}+\frac{\left(x+8\right)}{\left(x+8\right)}×\frac{4}{\left(x-3\right)}$Multiply the second ratio to obtain a common denominator.

$\frac{2x+4\left(x+8\right)}{\left(x+8\right)\left(x-3\right)}$Add the numerators.

$\frac{2x+4x+32}{\left(x+8\right)\left(x-3\right)}$Use the distributive property to combine like terms.

$\frac{6x+32}{\left(x+8\right)\left(x-3\right)}$Rewrite in simplified form.

Example 5: Special products in the denominator.

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

x2 - 4 = (x + 2)(x - 2) and x2 + 4x + 4 = (x + 2)(x + 2)Factor the denominators.

The LCM is (x + 2)(x + 2)(x - 2).

Multiply the ratios to obtain common denominators.

$\frac{x\left(x+2\right)}{\left(x+2\right)\left(x+2\right)\left(x-2\right)}+\frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)\left(x+2\right)}$Rewrite.

$\frac{x\left(x+2\right)+\left(x-2\right)\left(x+2\right)}{\left(x+2\right)\left(x+2\right)\left(x-2\right)}$Add the numerators.

$\frac{\left(x+\left(x-2\right)\right)\left(x+2\right)}{\left(x+2\right)\left(x+2\right)\left(x-2\right)}$   Regroup the terms to factor the numerator.

$\frac{\left(2x-2\right)\overline{){\left(}{x}{+}{2}{\right)}}}{\left(x+2\right)\overline{){\left(}{x}{+}{2}{\right)}}\left(x-2\right)}$  Eliminate common factors.

$\frac{\left(2x-2\right)}{\left(x+2\right)\left(x-2\right)}$    Rewrite in simplified form.

Example 6: Subtraction
$\frac{x+3}{{x}^{2}-2x-8}-\frac{x-5}{{x}^{2}-12x+32}$

x2 - 2x - 8 = (x - 4)(x + 2)Factor the denominators.

The LCM is (x - 8)(x - 4)(x + 2).

$\frac{\left(x-8\right)}{\left(x-8\right)}×\frac{\left(x+3\right)}{\left(x-4\right)\left(x+2\right)}-\frac{\left(x+2\right)}{\left(x+2\right)}×\frac{\left(x-5\right)}{\left(x-8\right)\left(x-4\right)}$   Multiply the ratios to obtain common denominators.

$\frac{{x}^{2}-5x-24}{\left(x-8\right)\left(x-4\right)\left(x+2\right)}-\frac{{x}^{2}-3x-10}{\left(x-8\right)\left(x-4\right)\left(x+2\right)}$Multiply to determine new numerators

$\frac{-2x-14}{\left(x-8\right)\left(x-4\right)\left(x+2\right)}$        Subtract the second numerator from the first.

With -2x - 14 = -2(x + 7)    there are no common factors to simplify the ratio; keep the ratio as is.

Example 7: Addition and subtraction together

$\frac{x+3}{{x}^{2}-25}+\frac{x-1}{x-5}-\frac{2}{x+5}$

x2 - 25 = (x + 5)(x - 5)Factor the denominator.

The LCM is (x + 5)(x - 5).

$\frac{\left(x+3\right)}{\left(x-5\right)\left(x+5\right)}+\frac{\left(x+5\right)}{\left(x+5\right)}×\frac{\left(x-1\right)}{\left(x-5\right)}-\frac{\left(x-5\right)}{\left(x-5\right)}×\frac{2}{\left(x+5\right)}$     Multiply the ratios to obtain a common denominator.

Rewrite the numerator.

$\frac{{x}^{2}+3x+8}{\left(x+3\right)\left(x-5\right)\left(x+5\right)}$          Combine like terms.

Now that you can add and subtract rational expressions, you are ready to start solving rational equations.

 Related Links: Math algebra Adding and Subtracting Rational Expressions Worksheets Rational Expressions Multiplying Rational Equations Simplifying Rational Expressions Algebra Topics

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