If 3+ |n^2 - 2| ≤ 12, then what is the number of integer values of n w

Question

If 3+ |n^2 - 2| ≤ 12, then what is the number of integer values of n which satisfy the inequality?

(A) 3

(B) 4

(C) 5

(D) 7

(E) 9

TestPrepUnlimited · Accepted Answer

We can simplify the inequality first,

3+ |n^2 - 2| ≤ 12

|n^2 - 2| ≤ 9

-9 ≤ n^2 - 2 ≤ 9

-7 ≤ n^2 ≤ 11

Since a square cannot be negative we can ignore the left side. We want to know how many integers satisfy  n^2 ≤ 11 , which is all the integers from -3 to 3. Hence seven integers including 0.

Ans: D

shameekv1989 · Answer

3+ |n^2 - 2| ≤ 12 -> |n^2 - 2| ≤ 9

When  n^2 - 2 ≤ 0  i.e.  0 ≤ n^2 ≤ 2 , 
 n^{2} - 2   \geq {-9}  i.e., 
 n^2   \geq {-7}  i.e. 
0 ≤  n^2  ≤ 2 i.e., 
0 ≤ |n| ≤  \sqrt{2} 
Therefore n = 0,1,-1

When  n^{2} - 2 \geq{0}  i.e.  n^2 \geq {2} , 
 n^2  - 2 ≤ 9 i.e., 
 n^2  - 2 ≤ 11 i.e., 
2 ≤  n^2  ≤ 11 i.e.,
 \sqrt{2}  ≤ |n| ≤  \sqrt{11} 
Therefore n = 2,-2; 3,-3

Total integer values = 7 - Answer D

shameekv1989 · Answer

U can square both sides to get n^4 -4n^2 <= 77

And therefore you have n^2 (n+2) (n-2) <= 77

Try 4 you will get 12*16 > 77
Try 3 you will get 5*9 < 77

Since we have n^2 and n^4 terms we can say +- values both will satisfy this condition.

Therefore you have all the integer values from -3 to +3 = 7 integer values

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BrushMyQuant · Answer

3+ | n^2  - 2| ≤ 12 and we need to find what is the number of integer values of n which satisfy the inequality?

3+ | n^2  - 2| ≤ 12 
=> | n^2  - 2| ≤ 9

We will have two cases when we open the Absolute Value   
 
   -
    Case 1:   n^2  - 2 ≥ 0 or  n^2  ≥ 2
  
=> | n^2  - 2| =  n^2  - 2
=>  n^2  - 2 ≤ 9
 =>  n^2  ≤ 11

But condition was    n^2  ≥ 2
 => 2 ≤  n^2  ≤ 11
  => Possible integer values of n are -3, -2, 2, 3  
  -
    Case 2:   n^2  - 2 ≤ 0 or  n^2  ≤ 2
  
 => | n^2  - 2| = -( n^2  - 2)
=> -( n^2  - 2) ≤ 9
=>  n^2  ≥ 2- 9
 =>  n^2  ≥ -7

But  n^2  will always be ≥ 0 and the condition was  n^2  ≤ 2   
=> 0 ≤  n^2  ≤ 2
  => Possible integer values of n are -2, -1, 0, 1, 2

=> Possible values of n are -3, -2, -1, 0, 1, 2, 3
=> 7 values

So,  Answer will be D 
Hope it helps!

Watch the following video to MASTER Absolute Values

https://www.youtube.com/watch?v=huFiBk8PaGY

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If 3+ |n^2 - 2| ≤ 12, then what is the number of integer values of n w

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We will have two cases when we open the Absolute Value
- Case 1: \(n^2\) - 2 ≥ 0 or \(n^2\) ≥ 2 => \|\(n^2\) - 2\| = \(n^2\) - 2 => \(n^2\) - 2 ≤ 9 => \(n^2\) ≤ 11 But condition was \(n^2\) ≥ 2 => 2 ≤ \(n^2\) ≤ 11 => Possible integer values of n are -3, -2, 2, 3	- Case 2: \(n^2\) - 2 ≤ 0 or \(n^2\) ≤ 2 => \|\(n^2\) - 2\| = -(\(n^2\) - 2) => -(\(n^2\) - 2) ≤ 9 => \(n^2\) ≥ 2- 9 => \(n^2\) ≥ -7 But \(n^2\) will always be ≥ 0 and the condition was \(n^2\) ≤ 2 => 0 ≤ \(n^2\) ≤ 2 => Possible integer values of n are -2, -1, 0, 1, 2

GMAT Problem Solving (PS) Questions

If 3+ |n^2 - 2| ≤ 12, then what is the number of integer values of n w

Prep Toolkit

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