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26 Apr 2023, 05:21
Kudos
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26 Apr 2023, 05:21
Kudos
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Official Solution:
Is \(x^2 < x^3\)?
We can simplify the question by dividing the inequality by \(x^2\) to get: is \(1 < x\)? We can safely do this because \(x^2\) cannot be negative. Also, note that \(x = 0\) does not satisfy either \(x^2 < x^3\) or the rephrased version \(1 < x\). Therefore, by dividing by \(x^2\), we are not introducing the possibility of \(x\) being 0 for the rephrased version, which ensures the validity of this operation.
(1) \(x < x^2\)
Rewrite the inequality as \(x^2 - x > 0\), and then as \(x(x - 1) > 0\). The roots are \(x=0\) and \(x = 1\). The "\(>\)" sign indicates that the solution lies to the left of the smaller root and to the right of the larger root; therefore, the given inequality holds true for: \(x < 0\) and \(x > 1\). Hence, \(x\) may or may not be greater than 1. Not sufficient.
(2) \(x < 1\)
This statement directly provides a NO answer to the question. Sufficient.
Answer: B
Is \(x^2 < x^3\)?
We can simplify the question by dividing the inequality by \(x^2\) to get: is \(1 < x\)? We can safely do this because \(x^2\) cannot be negative. Also, note that \(x = 0\) does not satisfy either \(x^2 < x^3\) or the rephrased version \(1 < x\). Therefore, by dividing by \(x^2\), we are not introducing the possibility of \(x\) being 0 for the rephrased version, which ensures the validity of this operation.
(1) \(x < x^2\)
Rewrite the inequality as \(x^2 - x > 0\), and then as \(x(x - 1) > 0\). The roots are \(x=0\) and \(x = 1\). The "\(>\)" sign indicates that the solution lies to the left of the smaller root and to the right of the larger root; therefore, the given inequality holds true for: \(x < 0\) and \(x > 1\). Hence, \(x\) may or may not be greater than 1. Not sufficient.
(2) \(x < 1\)
This statement directly provides a NO answer to the question. Sufficient.
Answer: B
29 Apr 2023, 07:14
Kudos
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2 doubts :
1) How can we divide the equation by 0; what if x is 0- then x^2 is 0 and hence the equation will be undefined
2) For statement 2)- if x=-1/2, then x^2 < x^3 (ANSWER will be YES and not NO right?)
1) How can we divide the equation by 0; what if x is 0- then x^2 is 0 and hence the equation will be undefined
2) For statement 2)- if x=-1/2, then x^2 < x^3 (ANSWER will be YES and not NO right?)
