System of Absolute Value Inequalities

We have two inequalities:

1. $|n - 60| < |n - 100|$ Squaring both sides: $(n - 60)^2 < (n - 100)^2 \implies n^2 - 120n + 3600 < n^2 - 200n + 10000$ $80n < 6400 \implies n < 80$

2. $|n - 100| < |n - 20|$ Squaring both sides: $(n - 100)^2 < (n - 20)^2 \implies n^2 - 200n + 10000 < n^2 - 40n + 400$ $9600 < 160n \implies n > 60$

Combining both inequalities gives $60 < n < 80$. The integer values of $n$ are $61, 62, \dots, 79$. The total number of integers is $79 - 61 + 1 = 19$.