Before diving into the solution, let's understand what these absolute value expressions represent: $|n-60|$ = distance from point $n$ to point $60$ on the number line $|n-100|$ = distance from point $n$ to point $100$ on the number line $|n-20|$ = distance from point $n$ to point $20$ on the number line Think of it this way: If you're standing at position $n$ on a number line, these expressions tell you how far you are from the points 60, 100, and 20 respectively.

We need to solve: $|n-60| < |n-100| < |n-20|$ This gives us two separate conditions: 1. $|n-60| < |n-100|$ (n is closer to 60 than to 100) 2. $|n-100| < |n-20|$ (n is closer to 100 than to 20)

When is a point closer to 60 than to 100? The answer lies in finding the midpoint between 60 and 100: $\text{Midpoint} = \frac{60 + 100}{2} = \frac{160}{2} = 80$ Any point to the left of 80 is closer to 60, and any point to the right of 80 is closer to 100. Let's verify: At $n = 80$: $|80-60| = 20$ and $|80-100| = 20$ (equal distances) At $n = 79$: $|79-60| = 19$ and $|79-100| = 21$ (closer to 60) At $n = 81$: $|81-60| = 21$ and $|81-100| = 19$ (closer to 100) Therefore: $|n-60| < |n-100|$ when $n < 80$

When is a point closer to 100 than to 20? We find the midpoint between 100 and 20: $\text{Midpoint} = \frac{100 + 20}{2} = \frac{120}{2} = 60$ Let's verify: At $n = 60$: $|60-100| = 40$ and $|60-20| = 40$ (equal distances) At $n = 61$: $|61-100| = 39$ and $|61-20| = 41$ (closer to 100) At $n = 59$: $|59-100| = 41$ and $|59-20| = 39$ (closer to 20) Therefore: $|n-100| < |n-20|$ when $n > 60$

From our analysis: Condition 1: $n < 80$ Condition 2: $n > 60$ Combined: $60 < n < 80$

The integers satisfying $60 < n < 80$ are: $n \in \{61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79\}$ Counting: From 61 to 79 inclusive = $79 - 61 + 1 = 19$ integers

Let's verify with $n = 70$: $|70-60| = 10$ $|70-100| = 30$ $|70-20| = 50$ Check: $10 < 30 < 50$ Answer: There are 19 integers that satisfy the given inequalities.

When comparing distances using absolute values, the midpoint between two reference points is the boundary where distances become equal. This technique works for any similar absolute value inequality problem.

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