When we add or subtract rational expressions, we follow the same procedures we used with fractions. To add or subtract rational expressions, we must first obtain a common denominator. We then add or subtract numerators and place the result over the common denominator. Lastly, we factor numerator and denominator, cancel any common factors, and report a simplified answer.

Test Objectives
• Demonstrate the ability to find the LCD for a group of rational expressions
• Demonstrate the ability to add rational expressions
• Demonstrate the ability to subtract rational expressions
Adding and Subtracting Rational Expressions Practice Test:

#1:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{6x}{x + 1}- \frac{5}{5x^3}$$

$$b)\hspace{.2em}\frac{5}{2x - 7}+ \frac{6}{3x - 5}$$

#2:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{6}{x^2 - 3x - 10}- \frac{6}{7x^2}$$

$$b)\hspace{.2em}\frac{5x}{8x - 7}+ \frac{5}{2x - 6}$$

#3:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{3x}{x - 6}+ \frac{6x}{x - 5}$$

$$b)\hspace{.2em}\frac{3x + 1}{x - 6}+ \frac{8}{8x - 32}$$

#4:

Instructions: Simplify each.

$$a)\hspace{.2em}\frac{7x}{x + 5}+ \frac{6}{x + 9}$$

$$b)\hspace{.2em}\frac{12x}{x - 12}- \frac{11}{11x + 11}$$

#5:

Instructions: Simplify each.

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$$a)\hspace{.2em}\frac{x^2 - 14x + 33}{x^2 - 6x - 55}- \frac{5}{3x^2 + 15x}$$

$$b)\hspace{.2em}\frac{5x + 10}{8x^4 + 20x^3 + 8x^2}+ \frac{x^2 + 3x}{8x^5 + 20x^4 - 12x^3}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\frac{6x^4 - x - 1}{x^3(x + 1)}$$

$$b)\hspace{.2em}\frac{27x - 67}{(2x - 7)(3x - 5)}$$

#2:

Solutions:

$$a)\hspace{.2em}\frac{6(6x^2 + 3x + 10)}{7x^2(x + 2)(x - 5)}$$

$$b)\hspace{.2em}\frac{5(2x^2 + 2x - 7)}{2(8x - 7)(x - 3)}$$

#3:

Solutions:

$$a)\hspace{.2em}\frac{3x(3x - 17)}{(x - 5)(x - 6)}$$

$$b)\hspace{.2em}\frac{3x^2 - 10x - 10}{(x - 4)(x - 6)}$$

#4:

Solutions:

$$a)\hspace{.2em}\frac{7x^2 + 69x + 30}{(x + 5)(x + 9)}$$

$$b)\hspace{.2em}\frac{12x^2 + 11x + 12}{(x + 1)(x - 12)}$$

#5:

Solutions:

$$a)\hspace{.2em}\frac{3x^2 - 9x - 5}{3x(x + 5)}$$

$$b)\hspace{.2em}\frac{3x - 1}{x^2(2x + 1)(2x - 1)}$$