Lesson Objectives

- Demonstrate an understanding of how to add and subtract fractions
- Learn how to find the LCD for a group of rational expressions
- Learn how to add rational expressions
- Learn how to subtract rational expressions

## How to Add & Subtract Rational Expressions

When we add or subtract rational expressions, we follow the same rules we used with fractions. When we add or subtract rational expressions with a common denominator, we can add or subtract numerators and place the result over the common denominator. Once this is done, we want to simplify our answer. Let's look at an example.

Example 1: Perform each indicated operation. $$\frac{3x + 5}{16x}+ \frac{x + 3}{16x}$$ Since the denominators are the same, we can just add the numerators: $$\frac{3x + 5 + x + 3}{16x}$$ $$\frac{4x + 8}{16x}$$ Simplify: $$\require{cancel}\frac{4(x + 2)}{16x}$$ $$\frac{\cancel{4}(x + 2)}{4\cancel{16}x}$$ $$\frac{x + 2}{4x}$$

Example 2: Find the LCD. $$\frac{x - 1}{x^2 + 10x + 25}, \frac{2x - 3}{x^2 + 7x + 10}, \frac{x^3 - 13}{x^2 + 11x + 18}$$ Step 1) Factor each denominator: $$x^2 + 10x + 25=(x + 5)(x + 5)$$ $$x^2 + 7x + 10=(x + 5)(x + 2)$$ $$x^2 + 11x + 18=(x + 2)(x + 9)$$ Step 2) Find the LCD: $$(x + 5)^2 (x + 2)(x + 9)$$

Example 3: Perform each indicated operation. $$\frac{x + 4}{x - 7}+ \frac{x - 8}{x - 3}$$ Step 1) Find the LCD of the rational expressions:

LCD: $$(x - 7)(x - 3)$$ Step 2) Transform each rational expression into an equivalent rational expression with the LCD as its denominator. $$\frac{x + 4}{x - 7}\cdot \frac{x - 3}{x - 3}=\frac{(x + 4)(x - 3)}{(x - 7)(x - 3)}$$ $$\frac{x - 8}{x - 3}\cdot \frac{x - 7}{x - 7}=\frac{(x - 8)(x - 7)}{(x - 3)(x - 7)}$$ Step 3) Perform the operations with the numerators: $$\frac{(x + 4)(x - 3)}{(x - 7)(x - 3)}+ \frac{(x - 8)(x - 7)}{(x - 3)(x - 7)}$$ $$\frac{x^2 + x - 12 + x^2 - 15x + 56}{(x - 3)(x - 7)}$$ $$\frac{2x^2 - 14x + 44}{(x - 3)(x - 7)}$$ Step 4) Simplify: $$\frac{2(x^2 - 7x + 22)}{(x - 3)(x - 7)}$$ Since there are no common factors that can be canceled, we can report our answer. In many cases, it is preferable to leave the rational expression in factored form. $$\frac{2(x^2 - 7x + 22)}{(x - 3)(x - 7)}$$ Example 4: Perform each indicated operation. $$\frac{5x - y}{x^2 + xy - 2y^2}- \frac{3x + 2y}{x^2 + 5xy - 6y^2}$$ Step 1) Find the LCD of the rational expressions:

$$\frac{5x - y}{(x - y)(x + 2y)}- \frac{3x + 2y}{(x - y)(x + 6y)}$$ LCD: $$(x - y)(x + 2y)(x + 6y)$$ Step 2) Transform each rational expression into an equivalent rational expression with the LCD as its denominator. $$\frac{5x - y}{(x - y)(x + 2y)}\cdot \frac{x + 6y}{x + 6y}=\frac{(5x - y)(x + 6y)}{(x - y)(x + 2y)(x + 6y)}=\frac{5x^2 + 29xy - 6y^2}{(x - y)(x + 2y)(x + 6y)}$$ $$\frac{3x + 2y}{(x - y)(x + 6y)}\cdot \frac{x + 2y}{x + 2y}=\frac{(3x + 2y)(x + 2y)}{(x - y)(x + 6y)(x + 2y)}=\frac{3x^2 + 8xy + 4y^2}{(x - y)(x + 6y)(x + 2y)}$$ Step 3) Perform the operations with the numerators: $$\frac{5x^2 + 29xy - 6y^2}{(x - y)(x + 2y)(x + 6y)}- \frac{3x^2 + 8xy + 4y^2}{(x - y)(x + 2y)(x + 6y)}$$ $$\frac{5x^2 + 29xy - 6y^2}{(x - y)(x + 2y)(x + 6y)}+ \frac{-3x^2 - 8xy - 4y^2}{(x - y)(x + 2y)(x + 6y)}$$ $$\frac{5x^2 + 29xy - 6y^2 - 3x^2 - 8xy - 4y^2}{(x - y)(x + 2y)(x + 6y)}$$ $$\frac{2x^2 + 21xy - 10y^2}{(x - y)(x + 2y)(x + 6y)}$$ Step 4) Simplify:

Our numerator is prime, therefore we can't simplify this rational expression any further. We will report our answer as: $$\frac{2x^2 + 21xy - 10y^2}{(x - y)(x + 2y)(x + 6y)}$$

Example 1: Perform each indicated operation. $$\frac{3x + 5}{16x}+ \frac{x + 3}{16x}$$ Since the denominators are the same, we can just add the numerators: $$\frac{3x + 5 + x + 3}{16x}$$ $$\frac{4x + 8}{16x}$$ Simplify: $$\require{cancel}\frac{4(x + 2)}{16x}$$ $$\frac{\cancel{4}(x + 2)}{4\cancel{16}x}$$ $$\frac{x + 2}{4x}$$

### Finding the LCD for a group of rational expressions

- Factor each denominator
- The LCD is the product of all different factors from each of the denominators. Each factor is raised to the largest power that occurs in any of the denominators

Example 2: Find the LCD. $$\frac{x - 1}{x^2 + 10x + 25}, \frac{2x - 3}{x^2 + 7x + 10}, \frac{x^3 - 13}{x^2 + 11x + 18}$$ Step 1) Factor each denominator: $$x^2 + 10x + 25=(x + 5)(x + 5)$$ $$x^2 + 7x + 10=(x + 5)(x + 2)$$ $$x^2 + 11x + 18=(x + 2)(x + 9)$$ Step 2) Find the LCD: $$(x + 5)^2 (x + 2)(x + 9)$$

### Adding & Subtracting Rational Expressions Without a Common Denominator

When we add or subtract rational expressions, we may not have a common denominator. When this occurs, we will first find the LCD of all rational expressions. We can then transform each rational expression into an equivalent rational expression with the LCD as its denominator. Once this is done, we perform our operations with the numerators and then simplify the answer. Let's look at a few examples.Example 3: Perform each indicated operation. $$\frac{x + 4}{x - 7}+ \frac{x - 8}{x - 3}$$ Step 1) Find the LCD of the rational expressions:

LCD: $$(x - 7)(x - 3)$$ Step 2) Transform each rational expression into an equivalent rational expression with the LCD as its denominator. $$\frac{x + 4}{x - 7}\cdot \frac{x - 3}{x - 3}=\frac{(x + 4)(x - 3)}{(x - 7)(x - 3)}$$ $$\frac{x - 8}{x - 3}\cdot \frac{x - 7}{x - 7}=\frac{(x - 8)(x - 7)}{(x - 3)(x - 7)}$$ Step 3) Perform the operations with the numerators: $$\frac{(x + 4)(x - 3)}{(x - 7)(x - 3)}+ \frac{(x - 8)(x - 7)}{(x - 3)(x - 7)}$$ $$\frac{x^2 + x - 12 + x^2 - 15x + 56}{(x - 3)(x - 7)}$$ $$\frac{2x^2 - 14x + 44}{(x - 3)(x - 7)}$$ Step 4) Simplify: $$\frac{2(x^2 - 7x + 22)}{(x - 3)(x - 7)}$$ Since there are no common factors that can be canceled, we can report our answer. In many cases, it is preferable to leave the rational expression in factored form. $$\frac{2(x^2 - 7x + 22)}{(x - 3)(x - 7)}$$ Example 4: Perform each indicated operation. $$\frac{5x - y}{x^2 + xy - 2y^2}- \frac{3x + 2y}{x^2 + 5xy - 6y^2}$$ Step 1) Find the LCD of the rational expressions:

$$\frac{5x - y}{(x - y)(x + 2y)}- \frac{3x + 2y}{(x - y)(x + 6y)}$$ LCD: $$(x - y)(x + 2y)(x + 6y)$$ Step 2) Transform each rational expression into an equivalent rational expression with the LCD as its denominator. $$\frac{5x - y}{(x - y)(x + 2y)}\cdot \frac{x + 6y}{x + 6y}=\frac{(5x - y)(x + 6y)}{(x - y)(x + 2y)(x + 6y)}=\frac{5x^2 + 29xy - 6y^2}{(x - y)(x + 2y)(x + 6y)}$$ $$\frac{3x + 2y}{(x - y)(x + 6y)}\cdot \frac{x + 2y}{x + 2y}=\frac{(3x + 2y)(x + 2y)}{(x - y)(x + 6y)(x + 2y)}=\frac{3x^2 + 8xy + 4y^2}{(x - y)(x + 6y)(x + 2y)}$$ Step 3) Perform the operations with the numerators: $$\frac{5x^2 + 29xy - 6y^2}{(x - y)(x + 2y)(x + 6y)}- \frac{3x^2 + 8xy + 4y^2}{(x - y)(x + 2y)(x + 6y)}$$ $$\frac{5x^2 + 29xy - 6y^2}{(x - y)(x + 2y)(x + 6y)}+ \frac{-3x^2 - 8xy - 4y^2}{(x - y)(x + 2y)(x + 6y)}$$ $$\frac{5x^2 + 29xy - 6y^2 - 3x^2 - 8xy - 4y^2}{(x - y)(x + 2y)(x + 6y)}$$ $$\frac{2x^2 + 21xy - 10y^2}{(x - y)(x + 2y)(x + 6y)}$$ Step 4) Simplify:

Our numerator is prime, therefore we can't simplify this rational expression any further. We will report our answer as: $$\frac{2x^2 + 21xy - 10y^2}{(x - y)(x + 2y)(x + 6y)}$$

#### Skills Check:

Example #1

Simplify each. $$\frac{9}{3x - 8}- \frac{5}{7x - 7}$$

Please choose the best answer.

A

$$\frac{7x^{2}+ 48x - 30}{7(x - 1)(3x - 8)}$$

B

$$\frac{48x - 23}{7(x - 1)(3x - 8)}$$

C

$$\frac{7x^{2}+ 41x - 23}{7(x - 1)(3x - 8)}$$

D

$$\frac{2x + 5}{x - 1}$$

E

$$\frac{4x + 2}{7(3x - 8)}$$

Example #2

Simplify each. $$\frac{8}{x + 6}- \frac{5}{x + 7}$$

Please choose the best answer.

A

$$\frac{3x + 26}{(x + 7)(x + 6)}$$

B

$$\frac{-7x - 2}{5(x + 6)}$$

C

$$\frac{15x - 90}{4(x + 4)}$$

D

$$\frac{3x}{2(x - 4)(x - 3)}$$

E

$$\frac{x}{x + 42}$$

Example #3

Simplify each. $$\frac{7}{x - 2}+ \frac{4x}{7x + 49}$$

Please choose the best answer.

A

$$\frac{36x}{7(x - 2)(x + 7)}$$

B

$$\frac{4x^{2}+ 48x + 392}{7(x - 2)(x + 7)}$$

C

$$\frac{36}{7(x - 2)}$$

D

$$\frac{4x^{2}+ 41x + 343}{7(x - 2)(x + 7)}$$

E

$$\frac{40x}{7(x - 2)(x + 7)}$$

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