Solution:
We have |n−60|<|n−100|<|n−20|
Now, the difference inside the modulus signified the distance of n from 60,100,and20 on the number line. This means that when the absolute difference from a number is larger, n would be further away from that number.
Example: The absolute difference of n and 60 is less than that of the absolute difference between n and 20. Hence, n cannot be , as then it would be closer to 20 than 60, and closer on the number line would indicate lesser value of absolute difference. Thus we have the condition that n>40
The absolute difference of n and 100 is less than that of the absolute difference between nand20 Hence, n cannot be , 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 nand100 Hence, n cannot be , 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 :
Dividing the range of n into 4 segments. $(n < 20, 20 100)$
1) For n<20
|n−20|=20−n,|n−60|=60−n,|n−100|=100−n
considering the inequality part :
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−nand100−n<n−20
For 100−n<n−20
120<2nandn>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−nand100−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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