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Question 18 of 18

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M:N:SPR:4

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SAT

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Passport to Advanced Math

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What is one possible solution to the equation fraction numerator 24 over denominator x plus 1 end fraction minus fraction numerator 12 over denominator x minus 1 end fraction equals 1 ?

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Correct Answer: 
No
Correct Answer: 
No
Correct Answer: 
No
Correct Answer: 
No

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5 or 7

Students should look for the best solution methods for solving rational equations before they begin. Looking for structure and common denominators will prove very useful at the onset, and will help prevent complex computations that do not lead to a solution.
 
In this problem, multiplying both sides of the equation by the common denominator open parentheses x plus 1 close parentheses open parentheses x minus 1 close parentheses yields 24 open parentheses x minus 1 close parentheses minus 12 left parenthesis x plus 1 right parenthesis equals left parenthesis x plus 1 right parenthesis left parenthesis x minus 1 right parenthesis.
 
Multiplication and simplification then yields 12 x minus 36 equals x squared minus 1 comma or x squared minus 12 x plus 35 equals 0. Factoring the quadratic gives open parentheses x minus 5 close parentheses open parentheses x minus 7 close parentheses equals 0 comma so the solutions occur at x equals 5 and x equals 7 comma both of which should be checked in the original equation to ensure that they are not extraneous. In this case, both values are solutions.
Question Difficulty: 
hard
Objective: 

Students must solve a rational equation in one variable.