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How many integer values of n satisfy the inequality

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How many integer values of n satisfy the inequality  [#permalink]

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New post 18 Feb 2019, 08:34
1
5
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A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

46% (01:33) correct 54% (01:26) wrong based on 115 sessions

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How many integer values of n satisfy the inequality \(3√n >√n^3\)?

A. 0

B. 1

C. 2

D. 4

E. Infinitely many

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How many integer values of n satisfy the inequality  [#permalink]

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New post 18 Feb 2019, 10:45
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Let's pick some numbers.
If n = 1, we have 3>1. Correct.
If n = 2, we have \(3\sqrt{2}>2\sqrt{2}\). Correct.
If n = 3, we have \(3\sqrt{3}=3\sqrt{3}\). Doesn't work.

We have understood that on the left side of the inequality we'll have \(3\sqrt{n}\) and on the right side \(n\sqrt{n}\). The inequality will only work for 1 and 2. We can't plug negative numbers because of the square root, so the answer is 2.

Answer C.
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Re: How many integer values of n satisfy the inequality  [#permalink]

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New post 18 Feb 2019, 19:10
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Solution



Given:
    • The number n is an integer

To find:
    • The number of values possible for n such that 3√n > √n3

Approach and Working:
Squaring both sides of the given inequality, we get
    • \(3√n > √n^3\)
    Or, \(9n > n^3\)
    Or, \(n^3 – 9n < 0\)
    Or, \(n (n^2 – 9) < 0\)

As n > 0, we can say \(n^2 – 9 < 0\), which implies
    • \(n^2 < 9\)
    Or, -3 < n < 3

But n cannot be negative.
Hence, range of possible values of n: 0 < n < 3
    • As n is integer, possible values of n = 1, 2

Hence, the correct answer is option C.

Answer: C

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How many integer values of n satisfy the inequality  [#permalink]

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New post 15 May 2019, 09:03
EgmatQuantExpert n<-3 would also satisfy the equation. Don't you think it should be E. I used the the wavy line method
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Re: How many integer values of n satisfy the inequality  [#permalink]

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New post 16 May 2019, 10:12
Why can the values be negative, i don't understand,

9n>n^3

try -8 & -4
they satisfy the equation.
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Re: How many integer values of n satisfy the inequality  [#permalink]

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New post 17 May 2019, 02:59
We can't consider negative numbers because in GMAT we don't work with square roots of negative numbers. It seems you're missing the square roots in the stem.
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Re: How many integer values of n satisfy the inequality  [#permalink]

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New post 21 May 2019, 13:37
Can this be done as?

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Re: How many integer values of n satisfy the inequality   [#permalink] 21 May 2019, 13:37
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