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How many distinct integer values of n for inequality | |n-3| +4| ≤ 12

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How many distinct integer values of n for inequality | |n-3| +4| ≤ 12 [#permalink]

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How many distinct integer values of n satisfy the inequality ||n-3| + 4| ≤ 12 ?

A. 15
B. 16
C. 17
D. 18
E. 19
[Reveal] Spoiler: OA

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How many distinct integer values of n for inequality | |n-3| +4| ≤ 12 [#permalink]

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GMATinsight wrote:
How many distinct integer values of n satisfy the inequality ||n-3| + 4| ≤ 12 ?

A. 15
B. 16
C. 17
D. 18
E. 19


METHOD-1

||n-3| + 4| ≤ 12 can be re-written as

i.e. -12 ≤ |n-3| + 4 ≤ 12

i.e. -12-4 ≤ |n-3| ≤ 12-4
i.e. -16 ≤ |n-3| ≤ 8

Here the corrective action needs to be taken.

-16 ≤ |n-3| in the above Inequation is REDUNTANT because we know that absolute value of any expression can't be less than zero. So an absolute value is greater than -16 doesn't contribute to our information at all

i.e. The Inequation can be sufficiently defined by the expression |n-3| ≤ 8

i.e. -8 ≤ n-3 ≤ 8
i.e. -8+3 ≤ n ≤ 8+3
i.e. -5 ≤ n ≤ 11

The Number of Integer values from -5 to 11 (Inclusive) is 17

Answer: Option C


METHOD-2

The primary expression is ||n-3| + 4| ≤ 12

i.e. The Absolute value of the expression on Left Hand Side (LHS) can go upto 12

i.e. The value of |n-3| can go upto 8

i.e. the Extreme absolute Value of n-3 can be +8

i.e. when n-3 = +8, The extreme value of n = +11
and when n-3 = -8, The extreme value of n = -5

i.e. The Value of n can range from -5 to +11 i.e. 17 Integer values

Answer: Option C
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How many distinct integer values of n for inequality | |n-3| +4| ≤ 12 [#permalink]

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New post 24 Feb 2016, 10:33
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GMATinsight wrote:
How many distinct integer values of n satisfy the inequality ||n-3| + 4| ≤ 12 ?

A. 15
B. 16
C. 17
D. 18
E. 19


can anyone tell me my approach....

considering LHS IIn-3I+4I is always +ve since In-3I and 4 both r +ve.
so i can write this as In-3I+4<=12
or In-3I<=8
so n can have -5 to 11 = 17 true values....

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Re: How many distinct integer values of n for inequality | |n-3| +4| ≤ 12 [#permalink]

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New post 27 Feb 2016, 20:29
rohit8865 wrote:
GMATinsight wrote:
How many distinct integer values of n satisfy the inequality ||n-3| + 4| ≤ 12 ?

A. 15
B. 16
C. 17
D. 18
E. 19


can anyone tell me my approach....

considering LHS IIn-3I+4I is always +ve since In-3I and 4 both r +ve.
so i can write this as In-3I+4<=12
or In-3I<=8
so n can have -5 to 11 = 17 true values....


Yes that's right...
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Re: How many distinct integer values of n for inequality | |n-3| +4| ≤ 12 [#permalink]

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New post 22 Mar 2016, 23:38
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the best way to solve is by solving the outer modules and the inner one.

||n-3|+4|=12
-12<=|n-3|+4<=12
-12-4<=|n-3|<=12-4 (subtract 4 from both sides)
-16<=|n-3|<=8

As the value of |n-3| can never be negative, so ignoring the -16 value.

-8<=|n-3|<=8
-8+3<=n<=8+3
-5<=n<=11

resulting in 17 distinct values.

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Re: How many distinct integer values of n for inequality | |n-3| +4| ≤ 12 [#permalink]

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Re: How many distinct integer values of n for inequality | |n-3| +4| ≤ 12 [#permalink]

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New post 09 Aug 2017, 09:02
GMATinsight wrote:
GMATinsight wrote:
How many distinct integer values of n satisfy the inequality ||n-3| + 4| ≤ 12 ?

A. 15
B. 16
C. 17
D. 18
E. 19


METHOD-1

||n-3| + 4| ≤ 12 can be re-written as

i.e. -12 ≤ |n-3| + 4 ≤ 12

i.e. -12-4 ≤ |n-3| ≤ 12-4
i.e. -16 ≤ |n-3| ≤ 8

Here the corrective action needs to be taken.

-16 ≤ |n-3| in the above Inequation is REDUNTANT because we know that absolute value of any expression can't be less than zero. So an absolute value is greater than -16 doesn't contribute to our information at all

i.e. The Inequation can be sufficiently defined by the expression |n-3| ≤ 8

i.e. -8 ≤ n-3 ≤ 8
i.e. -8+3 ≤ n ≤ 8+3
i.e. -5 ≤ n ≤ 11

The Number of Integer values from -5 to 11 (Inclusive) is 17

Answer: Option C


METHOD-2

The primary expression is ||n-3| + 4| ≤ 12

i.e. The Absolute value of the expression on Left Hand Side (LHS) can go upto 12

i.e. The value of |n-3| can go upto 8

i.e. the Extreme absolute Value of n-3 can be +8

i.e. when n-3 = +8, The extreme value of n = +11
and when n-3 = -8, The extreme value of n = -5

i.e. The Value of n can range from -5 to +11 i.e. 17 Integer values

Answer: Option C



hi

I have solved some problems of this kind previously, without any corrective action taken as such. I think the double absolute modulus has made this difference. Do you think, however, we are better off turning the double modulus into one, before setting the final inequality....? For example ...

|n-3| + 4| ≤ 12

| n-3 | <= 8

now we can set the inequality as under:

-8 <| n-3 | <= 8

OR, if there is any guiding principle to follow when solving questions as such, please say to me ...

thanks in advance ...

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Re: How many distinct integer values of n for inequality | |n-3| +4| ≤ 12   [#permalink] 09 Aug 2017, 09:02
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