There are three parts but its over Extraneous Solutions

Question

Part A Solve the following equation:  1/2x+2=1/x^2-1 . Show all work

Part B) Test the solution(s) found in Part A using substitution. Identify each solution as either valid or extraneous.

Part C)  Explain how the solution(s) of the equation 1/2x+2=1/x^2-1 can be tested using a graph.  

Answer 1

I'll assume the problems is

Cross multiply to get:

x² - 1 = 2x + 2

x² - 2x - 3 = 0

(x - 3)(x + 1) = 0

x = 3 and x = -1

Test (x = 3):

1/(2•3+2) = 1/(3² - 1)

1/8 = 1/8 true, so x = 3 is a valid solution

Test (for x = -1):

1/(2•(-1)+2) = 1/((-1)² - 1)

1/0 = 1/0 not true since division by zero is undefined so x = -1 is not a solution (aka, it is extraneous).

To find solutions via graphing, graph y = 1/(2x + 2) and graph y = (x² - 1) and find the points of intersection. Doing so confirms that the only solution is x = 3.