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.
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
n and 20 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
n and 100 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<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 :
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.