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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?
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.
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.
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.
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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