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CLAT Exam  >  CLAT Questions  >  The number of integers n that satisfy the ine... Start Learning for Free
The number of integers n that satisfy the inequalities | n - 60| < n - 100| < |n - 20| is 
  • a)
    21
  • b)
    19
  • c)
    18
  • d)
    20
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The number of integers n that satisfy the inequalities | n - 60| < ...
We have |n - 60| < |n - 100| < |n - 20|
Now, the difference inside the modulus signified the distance of n from 60, 100, and 20 on the number line.
This means that when the absolute difference from a number is larger, n would be further away from that number.
The absolute difference of n and 100 is less than that of the absolute difference between n and 20.
Hence, n cannot be ≤ 60, as then it would be closer to 20 than 100. Thus we have the condition that n>60.
The absolute difference of n and 60 is less than that of the absolute difference between n and 100.
Hence, n cannot be ≥ 80, as then it would be closer to 100 than 60.
Thus we have the condition that n<80.
The number which satisfies the conditions are 61, 62, 63, 64……79. Thus, a total of 19 numbers.
Alternatively
as per the given condition: |n - 60| < |n - 100| < |n - 20|
Dividing the range of n into 4 segments. (n < 20, 20<n<60, 60<n<100, n > 100)
1) For n < 20.
|n-20| = 20-n, |n-60| = 60- n, |n-100| = 100-n
considering the inequality part: |n - 100| < n - 20|
100 -n < 20 -n,
No value of n satisfies this condition.
2) For 20 < n < 60.
|n-20| = n-20, |n-60| = 60- n, |n-100| = 100-n.
60- n < 100 – n and 100 – n < n – 20
For 100 -n < n – 20.
120 < 2n and n > 60. But for the considered range n is less than 60.
3) For 60 < n < 100
|n-20| = n-20, |n-60| = n-60, |n-100| = 100-n
n-60 < 100-n and 100-n < n-20.
For the first part 2n < 160 and for the second part 120 < 2n.
n takes values from 61 …………….79.
A total of 19 values
4) For n > 100
|n-20| = n-20, |n-60| = n-60, |n-100| = n-100
n-60 < n – 100.
No value of n in the given range satisfies the given inequality.
Hence a total of 19 values satisfy the inequality.
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Most Upvoted Answer
The number of integers n that satisfy the inequalities | n - 60| < ...
Analysis:
To solve the given inequalities, we need to consider the cases when the expressions inside the absolute value signs are positive and negative.

Case 1: n - 60 > n - 100 and n - 100 < n - 20
Solving n - 60 > n - 100 gives:
n - 60 > n - 100
-40 > -60
This is true for all n.
Solving n - 100 < n - 20 gives:
n - 100 < n - 20
-20 < 0
This is also true for all n.
Therefore, all integers satisfy the inequalities in this case.

Case 2: n - 60 > -(n - 100) and n - 100 < n - 20
Solving n - 60 > -(n - 100) gives:
n - 60 > -n + 100
2n > 160
n > 80
Solving n - 100 < n - 20 gives:
-100 < -20
This is not possible.
Therefore, no integers satisfy the inequalities in this case.

Conclusion:
Since all integers satisfy the inequalities in Case 1, the number of integers that satisfy the given inequalities is the same as the number of integers, which is |n - 60| < n - 100. This can be calculated as follows:
n - 60 < n - 100
-60 < -100
This is true for all n.
Hence, the correct answer is option (B) 19.
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The number of integers n that satisfy the inequalities | n - 60| < n - 100| < |n - 20| isa)21b)19c)18d)20Correct answer is option 'B'. Can you explain this answer?
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