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What is the solution of the equation (x-1)/(1-x) = 0?

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by Max@Math Revolution » Thu Oct 11, 2018 12:28 am

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[Math Revolution GMAT math practice question]

What is the solution of the equation (x-1)/(1-x) = 0?

A. -1
B. 0
C. 1
D. 0 or 1
E. The equation has no solution.
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by fskilnik@GMATH » Thu Oct 11, 2018 6:39 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the solution of the equation (x-1)/(1-x) = 0?

A. -1
B. 0
C. 1
D. 0 or 1
E. The equation has no solution.
Important conceptual question. Congrats, Max!

BEFORE any manipulation/calculation is done, the "mathematical entity" (in this case, the fraction) must EXIST.

Hence we must impose the denominator different from zero, i.e., x different from 1.

We know that any fraction given (if it exists) is zero if, and only if, its numerator is zero.

Only x=1 satisfies that (x-1 = 0 implies x=1).

From the fact x=1 is not allowed, the equation given has no solutions!

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Scott@TargetTestPrep » Sat Oct 13, 2018 5:48 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the solution of the equation (x-1)/(1-x) = 0?

A. -1
B. 0
C. 1
D. 0 or 1
E. The equation has no solution.
In order for a fraction to be equal to 0, its numerator must be 0. So x - 1 = 0. This gives us x = 1. However, since the denominator also equals 0 when x = 1, we have division by 0, which is not defined. Therefore, x can't be 1. In that case, there are no values of x that makes the fraction (x-1)/(1-x) equal to 0.

Alternate solution:

Recall that (a - b)/(b - a) = -1 so long as a ≠ b. Therefore, (x-1)/(1-x) = -1 so long as x ≠ 1. Since x can't be 1 (otherwise the fraction would be undefined). Therefore, (x-1)/(1-x) can never be 0 for any values of x.

Answer: E

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by Max@Math Revolution » Sun Oct 14, 2018 5:44 pm
=>

In order for a fraction to be zero, its numerator must be 0. This occurs when x = 1. However, if x = 1, then the denominator of this fraction is 0. Thus, 0 is an erroneous solution, and the equation has no solution.

Therefore, the answer is E.
Answer: E
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