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13 Feb 2023, 00:25
Kudos
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
52% (02:07) correct
48%
(02:02)
wrong
based on 433
sessions
History
Date
Time
Result
Not Attempted Yet
If x > y > z, which of the following could be true?
I. \(x^3 > y^3 > z^3\)
II. \(z > y^2 > x^3\)
III. \(x^3 > z^3 > y^3\)
A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
I. \(x^3 > y^3 > z^3\)
II. \(z > y^2 > x^3\)
III. \(x^3 > z^3 > y^3\)
A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
13 Feb 2023, 02:41
Kudos
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TBT
I. \(x^3 > y^3 > z^3\):
- We can ALWAYS raise an inequality to a positive odd power (the same for taking a positive odd root from an inequality). So, if \(x > y > z\), then \(x^3 > y^3 > z^3\), \(x^5 > y^5 > z^5\), \(x^7 > y^7 > z^7\), will always be true.
II. \(z > y^2 > x^3\):
- In this option we have reversed order when raised to positive powers, so we immediately should think of positive proper fractions (fractions from 0 to 1). For example, if x > y > z is 0.3 > 0.2 > 0.1, then z > y^2 > x^3, would be 0.1 > 0.2^2 > 0.3^3, which is correct.
III. \(x^3 > z^3 > y^3\).
- As discussed in I, \(x > y > z\) always implies that \(x^3 > y^3 > z^3\), so \(x^3 > z^3 > y^3\) cannot be correct.
Answer: C.
General Discussion
13 Feb 2023, 01:26
Kudos
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TBT
Given the relative position of x , y and z on the number line
----- z -------- y --------- x -----------
I. \(x ^ 3 > y ^3 > z ^{ 3}\)
Yes! When x, y and z are positive and are greater than 1
--- 1 --- z -------- y --------- x ----------- \(z^3\) ---------- \(y^3\) -------- \(x^3\) --------
II. \(z > y ^ 2 > x ^ 3\)
Yes! When x, y and z are positive and are less than 1
---0 --- \(x^3\) ------- \(y^2\) --------- z ---------- y -------- x -------- 1 ------------
III. \( x ^ 3 > z ^ 3 > y ^ 3\)
This is not possible.
\(x^3 > z^3\), suggests that
- x is greater than 1, in that case \(z^3\) will not be greater than \(y^3 \)
- x is less than 0, in that case \(z^3\) will not be greater than \(y^3 \)
Option C
