Algebra: Which of the following are true?

Question

Assume a > 0, b > 0 and 1 < a^x < b^x for all x < 0. Which of the following are true?

a. a < b < 1

b. 1 < a < b

c. a < 1 < b

d. b < a < 1

e. b < 1 < a

Explain how you got to the answer.

Answer 1

try a specific example

x=-1, a=1/4, b=1/2

then

1<1/(1/2)<1/(1/4)

1<2<4

1>b>a

or

a<b<1

which fits only 1st choice: a.

Answer 2

Since we have a negative exponent ( x < 0 ), then we have to get the reciprocal of the bases {a, b} to this exponent positive. example:

(¼)^-2= 4² = 16

(⅛)^-2 = 8² = 64

That means the smaller the value of the base a the greater power (a^x) it has and vice versa.

Since the condition is 1 < a^x < b^x , then b < a.

Therefore we eliminate choices (a), (b) and (c).

For a^x, and b^x to be greater than 1, a and b has to be between 0 and 1 exclusive. We can't have choice (e) because the base a there is greater than 1. Therefore, the final answer is choice (d).