Quote:
If x > y^2 > z^4, which of the following statements could be true?
I. x>y>z
II. z>y>x
III. x>z>y
A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and II
We are given that x > y^2 > z^4 and need to determine which statements must be true. Let’s test each Roman Numeral.
I. x > y > z
Notice that the order of arrangement of x, y, and z in the inequality x > y > z is the same as the order of arrangement of x, y^2, and z^4 in the inequality x > y^2 > z^4, so we want to test positive integers in this case.
x = 10
y = 3
z = 1
Notice that 10 > 3 > 1 for x > y > z AND 10 > 9 > 1 for x > y^2 > z^4.
We see that I could be true.
II. z > y > x
Notice that the order of arrangement of x, y, and z in the inequality z > y > x differs from the order of arrangement of x, y^2, and z^4 in the inequality x > y^2 > z^4, so we want to test positive proper fractions in this case. This is because we need to decrease the value of y and z to make them work within the given inequality.
x = 1/5
y = 1/3
z = 1/2
Notice that 1/2 > 1/3 > 1/5 for z > y > x AND 1/5 > 1/9 > 1/16 for x > y^2 > z^4.
We see that II could be true.
III. x > z > y
Notice that the order of arrangement of y and z in the inequality x > z > y differs from the order of arrangement of y^2 and z^4 in the inequality x > y^2 > z^4, so we once again want to test positive proper fractions. This is because we need to decrease the value of z to make it work within the given inequality (that is, we want to swap the order of z4 and y2 even if z > y).
x = 1/2
y = 1/4
z = 1/3
Notice that 1/2 > 1/3 > 1/4 for x > z > y AND 1/2 > 1/16 > 1/81 for x > y^2 > z^4.
We see that III could be true.
Answer: E
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