If x and y are positive, is x < 10 < y ?

jjjinapinch wrote:If x and y are positive, is x < 10 < y ?
(1) x < y and xy = 100
(2) x^2 < 100 < y^2

We are given that x and y are positive and we must determine whether x < 10 < y.

Statement One Alone:

x < y and xy = 100

From statement one we know that y is greater than x and xy = 100. We can isolate x in the equation xy = 100 and obtain:

x = 100/y

We now can substitute 100/y for x in the inequality x < y and then isolate y.

100/y < y

100 <y^2

√100<√(y^2)

10 < |y|

Since we are told that y is positive, |y| > 10 means y > 10. Since xy = 100 and y is greater than 10, then x must be less than 10. Thus, x < 10 < y. Statement one alone is sufficient to answer the question.

Statement Two Alone:

x^2 < 100 < y^2

Taking the square root gives us:

√(x^2) < √100 < √(y^2)

|x| < 10 < |y|

Since x and y are both positive, x < 10 < y. Statement two alone is also sufficient to answer the question.

Answer: D