Complex rational expressions contain a rational expression in the numerator, denominator, or both. We can simplify these problems using the LCD method. The LCD method tells us to first find the LCD of all smaller denominators. We then multiply the numerator and denominator of the complex rational expression by this LCD.

Test Objectives
• Demonstrate the ability to find the LCD for a group of rational expressions
• Demonstrate the ability to multiply two rational expressions
• Demonstrate the ability to simplify a rational expression
Complex Rational Expressions Practice Test:

#1:

Instructions: Simplify each using the LCD method.

a) $$\frac{\frac{2}{x\hspace{.5em} -\hspace{.5em} 1} + \frac{y\hspace{.5em} -\hspace{.5em} 3}{x\hspace{.5em} -\hspace{.5em} 1}}{\frac{x\hspace{.5em} -\hspace{.5em}1} {y\hspace{.5em} -\hspace{.5em} 3} - \frac{4}{y\hspace{.5em} -\hspace{.5em} 3}}$$

#2:

Instructions: Simplify each using the LCD method.

a) $$\frac{\frac{a\hspace{.5em} -\hspace{.5em} 1}{2b\hspace{.5em} + \hspace{.5em}6} + \frac{a\hspace{.5em} -\hspace{.5em} 1}{b\hspace{.5em} +\hspace{.5em} 3}} {\frac{9}{2} - \frac{a \hspace{.5em}- \hspace{.5em}1}{b\hspace{.5em} + \hspace{.5em}3}}$$

#3:

Instructions: Simplify each using the LCD method.

a) $$\frac{\frac{m\hspace{.5em} +\hspace{.5em} 3}{n\hspace{.5em} + \hspace{.5em}2} - \frac{m\hspace{.5em} + \hspace{.5em}3}{2}} {\frac{9}{4} + \frac{m\hspace{.5em} +\hspace{.5em} 3}{n\hspace{.5em} + \hspace{.5em}2}}$$

#4:

Instructions: Simplify each using the LCD method.

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

#5:

Instructions: Simplify each using the LCD method.

a) $$\frac{\frac{1}{2} - \frac{y\hspace{.5em} - \hspace{.5em}2}{4}}{\frac{x\hspace{.5em} - \hspace{.5em}2}{y \hspace{.5em}- \hspace{.5em}2}}$$

Written Solutions:

#1:

Solutions:

a) $$\frac{(y - 3)(y - 1)}{(x - 1)(x - 5)}$$

#2:

Solutions:

a) $$\frac{3(a - 1)}{(-2a + 9b + 29)}$$

#3:

Solutions:

a) $$\frac{-2n(m + 3)}{4m + 9n + 30}$$

#4:

Solutions:

a) $$\frac{3v(2v - 3)(v + 3)}{-(u + 5)(v + 6)(v - 6)}$$

#5:

Solutions:

a) $$\frac{-1(y - 2)(y - 4)}{4(x - 2)}$$