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16 Sep 2014, 01:52
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E
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Difficulty:
35%
(medium)
Question Stats:
71% (02:00) correct
29%
(02:34)
wrong
based on 41
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\(x\) is replaced by \(1 - x\) everywhere in the expression \(\frac{1}{x} - \frac{1}{1 - x}\), with \(x \neq 0\) and \(x \neq 1\). If the result is then multiplied by \(x^2 - x\), the outcome equals
A. \(x + 1\)
B. \(x - 1\)
C. \(1 - x^2\)
D. \(2x - 1\)
E. \(1 - 2x\)
A. \(x + 1\)
B. \(x - 1\)
C. \(1 - x^2\)
D. \(2x - 1\)
E. \(1 - 2x\)
16 Sep 2014, 01:52
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Official Solution:
\(x\) is replaced by \(1 - x\) everywhere in the expression \(\frac{1}{x} - \frac{1}{1 - x}\), with \(x \neq 0\) and \(x \neq 1\). If the result is then multiplied by \(x^2 - x\), the outcome equals
A. \(x + 1\)
B. \(x - 1\)
C. \(1 - x^2\)
D. \(2x - 1\)
E. \(1 - 2x\)
First, carry out the replacement. Literally replace every \(x\) in the expression with \(1 - x\), putting parentheses around the \(1 - x\) in order to preserve proper order of operations:
Original: \(\frac{1}{x} - \frac{1}{1 - x}\)
Replacement:
\(\frac{1}{(1 - x)} - \frac{1}{1 - (1 - x)}\)
Now simplify the second denominator: \((1 - (1 - x)) = (1 - 1 + x) = x\)
So the replacement expression becomes this:
\(\frac{1}{(1 - x)} - \frac{1}{x}\)
This should make sense. If we replace \(x\) by \(1 - x\), then it turns out that we are also replacing \(1 - x\) by \(x\) (since \(1 - (1 - x) = x\)). Thus, the denominators of the original expression are simply swapped.
Now we can either combine these fractions first (by finding a common denominator) or go ahead and multiply by \(x^2 - x,\) as we are instructed to. Let’s take the latter approach.
\(( \frac{1}{1 - x} - \frac{1}{x} ) (x^2 - x)[M] \\
Instead of FOILing this product right away, we should factor the expression \)x^2 - x\( first. If we do so, we will be able to cancel denominators quickly.\\
\\
\)x^2 - x\( factors into \)(x - 1)x\(. We can now rewrite the product:\\
\)( \frac{1}{1 - x} - \frac{1}{x}) (x - 1)x = \frac{(x - 1)x}{(1 - x)} - \frac{(x - 1)x}{x}[/M]
The second term, \(\frac{(x - 1)x}{x}\), becomes just \(x - 1\) after we cancel the \(x\)'s.
Since \((x - 1) = -(1 - x)\), we can rewrite the first term as \(\frac{-(1 - x)x}{(1 - x)}\) and then cancel the \((1 - x)\)’s, leaving \(-x\).
So, the final result is \(-x - (x - 1) = -x - x + 1 = 1 - 2x\). This is the answer.
Separately, since this is a Variables In Choices problem, we could instead pick a number and calculate a target. Since 0 and 1 are disallowed, let's pick \(x = 2\). We are told that \(x\) should be replaced by \(1 - x\), so we calculate \(1 - x = -1\) and put in -1 wherever \(x\) is in the original expression.
\(\frac{1}{x} - \frac{1}{(1 - x)} = \frac{1}{(-1)} - \frac{1}{(1 - (-1))} = -1 - \frac{1}{2} = -\frac{3}{2}\)
Now multiply this number by \(x^2 - x = 2^2 - 2 = 2\). We get -3 as our target number.
Finally, we plug \(x = 2\) into the answer choices and look for -3:
(A) \(x + 1 = 2 + 1 = 3\)
(B) \(x - 1 = 2 - 1 = 1\)
(C) \(1 - x^2 = 1 - 2^2 = -3\)
(D) \(2x - 1 = 2(2) - 1 = 3\)
(E) \(1 - 2x = 1 - 2(2) = -3\)
We can eliminate choices A, B, and D, but to choose between C and E, we would need to pick another number.
Answer: E
\(x\) is replaced by \(1 - x\) everywhere in the expression \(\frac{1}{x} - \frac{1}{1 - x}\), with \(x \neq 0\) and \(x \neq 1\). If the result is then multiplied by \(x^2 - x\), the outcome equals
A. \(x + 1\)
B. \(x - 1\)
C. \(1 - x^2\)
D. \(2x - 1\)
E. \(1 - 2x\)
First, carry out the replacement. Literally replace every \(x\) in the expression with \(1 - x\), putting parentheses around the \(1 - x\) in order to preserve proper order of operations:
Original: \(\frac{1}{x} - \frac{1}{1 - x}\)
Replacement:
\(\frac{1}{(1 - x)} - \frac{1}{1 - (1 - x)}\)
Now simplify the second denominator: \((1 - (1 - x)) = (1 - 1 + x) = x\)
So the replacement expression becomes this:
\(\frac{1}{(1 - x)} - \frac{1}{x}\)
This should make sense. If we replace \(x\) by \(1 - x\), then it turns out that we are also replacing \(1 - x\) by \(x\) (since \(1 - (1 - x) = x\)). Thus, the denominators of the original expression are simply swapped.
Now we can either combine these fractions first (by finding a common denominator) or go ahead and multiply by \(x^2 - x,\) as we are instructed to. Let’s take the latter approach.
\(( \frac{1}{1 - x} - \frac{1}{x} ) (x^2 - x)[M] \\
Instead of FOILing this product right away, we should factor the expression \)x^2 - x\( first. If we do so, we will be able to cancel denominators quickly.\\
\\
\)x^2 - x\( factors into \)(x - 1)x\(. We can now rewrite the product:\\
\)( \frac{1}{1 - x} - \frac{1}{x}) (x - 1)x = \frac{(x - 1)x}{(1 - x)} - \frac{(x - 1)x}{x}[/M]
The second term, \(\frac{(x - 1)x}{x}\), becomes just \(x - 1\) after we cancel the \(x\)'s.
Since \((x - 1) = -(1 - x)\), we can rewrite the first term as \(\frac{-(1 - x)x}{(1 - x)}\) and then cancel the \((1 - x)\)’s, leaving \(-x\).
So, the final result is \(-x - (x - 1) = -x - x + 1 = 1 - 2x\). This is the answer.
Separately, since this is a Variables In Choices problem, we could instead pick a number and calculate a target. Since 0 and 1 are disallowed, let's pick \(x = 2\). We are told that \(x\) should be replaced by \(1 - x\), so we calculate \(1 - x = -1\) and put in -1 wherever \(x\) is in the original expression.
\(\frac{1}{x} - \frac{1}{(1 - x)} = \frac{1}{(-1)} - \frac{1}{(1 - (-1))} = -1 - \frac{1}{2} = -\frac{3}{2}\)
Now multiply this number by \(x^2 - x = 2^2 - 2 = 2\). We get -3 as our target number.
Finally, we plug \(x = 2\) into the answer choices and look for -3:
(A) \(x + 1 = 2 + 1 = 3\)
(B) \(x - 1 = 2 - 1 = 1\)
(C) \(1 - x^2 = 1 - 2^2 = -3\)
(D) \(2x - 1 = 2(2) - 1 = 3\)
(E) \(1 - 2x = 1 - 2(2) = -3\)
We can eliminate choices A, B, and D, but to choose between C and E, we would need to pick another number.
Answer: E
31 Jan 2018, 11:37
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Found an easier way to solve this problem.
Assume (1-x) to be equivalent to a variable r.
Now the solution becomes 1/r - 1/x * x*-r
=> (x-r)/xr * (-rx) (cancel the numerator and denominator to get -1(x-r) = r-x
=> Now substitute r = 1-x
=> 1-x-x = 1-2x
Solution = E.
Assume (1-x) to be equivalent to a variable r.
Now the solution becomes 1/r - 1/x * x*-r
=> (x-r)/xr * (-rx) (cancel the numerator and denominator to get -1(x-r) = r-x
=> Now substitute r = 1-x
=> 1-x-x = 1-2x
Solution = E.
