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19 Mar 2020, 23:07
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Difficulty:
25%
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Question Stats:
65% (01:10) correct
35%
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wrong
based on 92
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If \(-1 < x < 0\), which of the following must be true?
I. \(x < x^2\)
II. \(x^2 < x^3\)
III. \(x < x^3\)
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
I. \(x < x^2\)
II. \(x^2 < x^3\)
III. \(x < x^3\)
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
20 Mar 2020, 02:28
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X is between -1 and 0
x^2 will be positive so, x^2 is greater than x and x^3 (both negative values)
thus I is true and II is false
if x is -0.1 x^3 is -0.001 thus x is less than x^3 III is true
when values between 0 and 1 are cubed or squared their numeric values decrease examples:(x,x^2,x^3):(0.2,0.04,0.008)(0.3,0.09,0.027)(0.5,0.25,0.125) , but since the value is negative as numeric value is decreasing the overall value with the negative sign will increase.
Answer is D
x^2 will be positive so, x^2 is greater than x and x^3 (both negative values)
thus I is true and II is false
if x is -0.1 x^3 is -0.001 thus x is less than x^3 III is true
when values between 0 and 1 are cubed or squared their numeric values decrease examples:(x,x^2,x^3):(0.2,0.04,0.008)(0.3,0.09,0.027)(0.5,0.25,0.125) , but since the value is negative as numeric value is decreasing the overall value with the negative sign will increase.
Answer is D
20 Mar 2020, 04:45
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From the inequality given in the question stem, we can infer that x is a negative proper fraction. Considering this, we need to understand that, while x and \(x^3\) will be negative, \(x^2\) will be positive.
Therefore, we can deal with the statements just on the basis of the signs without even having to deal with the magnitudes of the numbers.
Statement I says x<\(x^2\). This is definitely true since a negative value will always be lesser than a positive value. Statement I is definitely true.
Answer options B can be eliminated.
Statement II says \(x^2\)<\(x^3\). This is definitely false since a positive value can never be lesser than a negative value. Statement II is definitely false, not definitely true.
Answer options C and E can be eliminated.
Statement III says x<\(x^3\). Since x is a negative proper fraction, this is definitely true.
For example, if x = -\(\frac{1}{2}\), \(x^3\) = -\(\frac{1}{8}\). Note that -\(\frac{1}{2}\) is actually smaller than -\(\frac{1}{8}\) and not bigger.
Note that the range of values from -1 to 0 are mirror images of values from 0 to 1. If x=\(\frac{1}{2}\),\(x^3\) = \(\frac{1}{8}\) and \(x^3\)<x.
Therefore, since -1<x<0, x<\(x^3\). Statement III is definitely true.
Answer option A can be eliminated. The correct answer option is D.
Hope that helps!
Therefore, we can deal with the statements just on the basis of the signs without even having to deal with the magnitudes of the numbers.
Statement I says x<\(x^2\). This is definitely true since a negative value will always be lesser than a positive value. Statement I is definitely true.
Answer options B can be eliminated.
Statement II says \(x^2\)<\(x^3\). This is definitely false since a positive value can never be lesser than a negative value. Statement II is definitely false, not definitely true.
Answer options C and E can be eliminated.
Statement III says x<\(x^3\). Since x is a negative proper fraction, this is definitely true.
For example, if x = -\(\frac{1}{2}\), \(x^3\) = -\(\frac{1}{8}\). Note that -\(\frac{1}{2}\) is actually smaller than -\(\frac{1}{8}\) and not bigger.
Note that the range of values from -1 to 0 are mirror images of values from 0 to 1. If x=\(\frac{1}{2}\),\(x^3\) = \(\frac{1}{8}\) and \(x^3\)<x.
Therefore, since -1<x<0, x<\(x^3\). Statement III is definitely true.
Answer option A can be eliminated. The correct answer option is D.
Hope that helps!
