uptowngirl92 wrote:Source of the question is: GMAT Prep Test 1
If x is positive which of the following could be correct ordering of 1/x, 2x, x^2.
1) x^2 < 2x < 1/x
2) x^2 < 1/x < 2x
3) 2x < x^2 < 1/x
Choices are:
1) None
2) 1 Only
3) 3 Only
4) I and 2 only
5) 1, 2, 3
How to proceed without plugging in?I plugged in in the exam and got only choice 1 as correct and hence marke db which is wrong:(
Hybrid approach to make things smooth
"Could be true" Find one positive instance for each case. Proving that there are no positive instances requires algebraic way.
"must be true" Find one negative instance for each case. Proving that there are all positive instances requires algebraic way.
Lets focus on algebraic way.
1. x^2 < 2x < 1/x
x^2 - 2x < 0 --> (0,2)
2x < 1/x
x^2 - (1/2) < 0 --> (-1/sqrt(2), 1/sqrt(2))
Combining together: (0, 1/sqrt(2))
The domain of this inequality is the above.
2. x^2 < 1/x < 2x
x^3 < 1
domain: (-inf, 1)
and x^2 > 1/2
domain: (-inf, 1/sqrt(2)) U (1/sqrt(2), +inf)
Find the intersection of both domains: (-inf, 1/sqrt(2). Note that x is positive.
(0, 1/sqrt(2)) is the domain.
3. 2x < x^2 < 1/x
x^2 - 2x > 0
domain: (-inf, 0) U (2, +inf)
x^3 < 1
domain: (-inf, 1)
Intersection: (-inf, 0)
But x is positive.
Some key things to remember:
(x-a)(x-b) > 0, (-inf, a) U (b, +inf), assumin a < b
(x-a)(x-b) < 0 (a, b), assumin a < b
(x^3-a^3) > 0 (a, +inf)
(x^3+a^3) < 0 (-inf, a)
Being cool and taking the restctions (like x being +ve) into account help your test.
Smart numbers help you to get rid of some choices, thereby ending up with fewer choices. There, focus on algebraic approach.
