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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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18 Feb 2019, 08:34
2
00:00

Difficulty:

45% (medium)

Question Stats:

51% (01:18) correct 49% (01:06) wrong based on 44 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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Joined: 17 May 2018
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How many integer values of n satisfy the inequality  [#permalink]

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18 Feb 2019, 10:45
1
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.

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

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18 Feb 2019, 19:10

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.

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Re: How many integer values of n satisfy the inequality   [#permalink] 18 Feb 2019, 19:10
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