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16 Sep 2014, 01:45
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What is the value of \(x\)?
(1) \(|x|=-\frac{4}{x}\)
(2) \(x=-|\frac{4}{x}|\)
(1) \(|x|=-\frac{4}{x}\)
(2) \(x=-|\frac{4}{x}|\)
16 Sep 2014, 01:45
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Official Solution:
What is the value of \(x\)?
(1) \(|x|=-\frac{4}{x}\).
The left-hand side of the equation is an absolute value, so it must be non-negative. Thus, the right-hand side must also be non-negative: \(-\frac{4}{x}\geq{0}\). This means \(x\leq{0}\), but since \(x\) is in the denominator, it cannot be zero. Hence, \(x < 0\).
Next, if \(x < 0\), then \(|x|=-x\). So, \(|x|=-\frac{4}{x}\) becomes \(-x=-\frac{4}{x}\). From this, it follows that \(x^2=4\), so \(x=2\) or \(x=-2\). Discard the positive root because we know that \(x\) must be negative, and we are left with \(x=-2\). Sufficient.
(2) \(x=-|\frac{4}{x}|\). Re-arrange: \(|\frac{4}{x}|=-x\).
Similarly here, the left-hand side of the equation is an absolute value, so it must be non-negative. Thus, the right-hand side, \(-x\), must also be non-negative, implying \(-x\geq{0}\). Therefore, \(x\leq{0}\). However, since \(x\) is in the denominator, it cannot be zero, so \(x < 0\).
Next, if \(x < 0\), then \(|x|=-x\). So, \(|\frac{4}{x}|=-x\) becomes \(\frac{4}{-x}=-x\). Simplifying, we get \(x^2=4\), so \(x=2\) or \(x=-2\). Discard the positive root because we know that \(x\) must be negative, and we are left with \(x=-2\). Sufficient.
Answer: D
What is the value of \(x\)?
(1) \(|x|=-\frac{4}{x}\).
The left-hand side of the equation is an absolute value, so it must be non-negative. Thus, the right-hand side must also be non-negative: \(-\frac{4}{x}\geq{0}\). This means \(x\leq{0}\), but since \(x\) is in the denominator, it cannot be zero. Hence, \(x < 0\).
Next, if \(x < 0\), then \(|x|=-x\). So, \(|x|=-\frac{4}{x}\) becomes \(-x=-\frac{4}{x}\). From this, it follows that \(x^2=4\), so \(x=2\) or \(x=-2\). Discard the positive root because we know that \(x\) must be negative, and we are left with \(x=-2\). Sufficient.
(2) \(x=-|\frac{4}{x}|\). Re-arrange: \(|\frac{4}{x}|=-x\).
Similarly here, the left-hand side of the equation is an absolute value, so it must be non-negative. Thus, the right-hand side, \(-x\), must also be non-negative, implying \(-x\geq{0}\). Therefore, \(x\leq{0}\). However, since \(x\) is in the denominator, it cannot be zero, so \(x < 0\).
Next, if \(x < 0\), then \(|x|=-x\). So, \(|\frac{4}{x}|=-x\) becomes \(\frac{4}{-x}=-x\). Simplifying, we get \(x^2=4\), so \(x=2\) or \(x=-2\). Discard the positive root because we know that \(x\) must be negative, and we are left with \(x=-2\). Sufficient.
Answer: D
01 Aug 2016, 03:01
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I think this is a high-quality question and I agree with explanation. Great question but must be classified as 700 not 600
