sandy wrote:

If \(0 < a < \frac{1}{b}< 1\), then which of the following must be true?

(A) \(a^2 > a > b > b^2\)

(B) \(b > a > a^2 > b^2\)

(C) \(b^2 > a > a^2 > b\)

(D) \(b^2 > a^2 > b > a\)

(E) \(b^2 > b > a > a^2\)

Let's solve this by

testing valuesGIVEN: 0 < a < 1/b < 1

Since a is POSITIVE, we can safely multiply all parts of the inequality by a to get: 0 <

a² < a/b <

aSo, one thing that is definitely true is that

a² < aCheck the answer choices

(A)

a² > a > b > b² NO GOOD. ELIMINATE

(B) b >

a > a² > b² WORKS. KEEP

(C) b² >

a > a² > b WORKS. KEEP

(D) b² >

a² > b >

a NO GOOD. ELIMINATE

(E) b² > b >

a > a² WORKS. KEEP

GIVEN: 0 < a < 1/b < 1

Since b is POSITIVE, we can safely multiply all parts of the inequality by b to get: 0 < ab < 1 < b

Hmmm, this new inequality doesn't have b and b² (which is needed to check the answer choices)

So, let's take 0 < ab < 1 < b and multiply all parts by b to get: 0 < ab² <

b < b²Let's check the answer choices to see which ones adhere to the inequality

b < b²(B)

b > a > a² >

b² NO GOOD. ELIMINATE

(C)

b² > a > a² >

b WORKS. KEEP

(E)

b² > b > a > a² WORKS. KEEP

We're left with 2 answer choices:

(C) b² > a > a² > b

(E) b² > b > a > a²

Answer choice C says a > b and answer choice E says b > a

So, which is it?

Well, we know that a < 1/b

So, it COULD be the case that a = 1/3 and b = 2, which means b > a

So, the correct answer must be E

Cheers,

Brent

_________________

Brent Hanneson – Creator of greenlighttestprep.com

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