### About Solving Equations with Rational Expressions:

We solve rational equations using the same strategies we employed when solving linear equations with fractions. We begin by finding the LCD of all denominators. We then multiply both sides of the equation by the LCD to clear all denominators. When this is done, we solve the resulting equation and check for extraneous solutions.

Test Objectives

- Demonstrate the ability to find the LCD for a group of rational expressions
- Demonstrate the ability to solve an equation with rational expressions
- Demonstrate the ability to find and reject any extraneous solutions

#1:

Instructions: Solve each equation.

a) $$\frac{7x + 1}{x - 5} = \frac{6x - 3}{x^2 - 25}$$

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#2:

Instructions: Solve each equation.

a) $$\frac{n + 4}{n + 5} = \frac{1}{n^2 + 2n - 15} + 1$$

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#3:

Instructions: Solve each equation.

a) $$\frac{4p + 24}{p^3 + 5p^2} - \frac{1}{p} = \frac{4}{p^3 + 5p^2}$$

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#4:

Instructions: Solve each equation.

a) $$1 = \frac{2}{p^2 -3p - 4} + \frac{p - 1}{p + 1}$$

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#5:

Instructions: Solve each equation.

a) $$\frac{5n}{n^3 - 4n} = \frac{-4}{n + 2} + \frac{1}{n - 2}$$

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Written Solutions:

#1:

Solutions:

a) $$x = -4\hspace{1em} or \hspace{1em} x = \frac{-2}{7}$$

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#2:

Solutions:

a) $$n = 2$$

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#3:

Solutions:

a) $$p = 4$$

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#4:

Solutions:

a) $$p = 5$$

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#5:

Solutions:

a) $$n = \frac{5}{3}$$