Question:medium

The number of integers \( n \) that satisfy the inequalities \( |n - 60| < |n - 100| < |n - 20| \) is 

Updated On: Jan 15, 2026
  • 21
  • 18
  • 20
  • 19
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The Correct Option is D

Solution and Explanation

The problem provides two inequalities:

  1. \( |n - 60| < |n - 100| \)
  2. \( |n - 100| < |n - 20| \)

Analysis of Inequality 1: \( |n - 60| < |n - 100| \)

We consider critical points where the expressions inside the absolute values are zero. These points are \(n=60\) and \(n=100\).

  • If \( n \le 60 \): \( -(n - 60) < -(n - 100) \implies -n + 60 < -n + 100 \implies 60 < 100 \). This is true for all \( n \le 60 \).
  • If \( 60 < n < 100 \): \( (n - 60) < -(n - 100) \implies n - 60 < -n + 100 \implies 2n < 160 \implies n < 80 \). Combined with the condition, this gives \( 60 < n < 80 \).
  • If \( n \ge 100 \): \( (n - 60) < (n - 100) \implies -60 < -100 \). This is false.

Therefore, the solution to the first inequality is \( n < 80 \). The integers in this range are \( \dots, 78, 79 \).

Analysis of Inequality 2: \( |n - 100| < |n - 20| \)

The critical points are \(n=20\) and \(n=100\).

  • If \( n \le 20 \): \( -(n - 100) < -(n - 20) \implies -n + 100 < -n + 20 \implies 100 < 20 \). This is false.
  • If \( 20 < n < 100 \): \( -(n - 100) < (n - 20) \implies -n + 100 < n - 20 \implies 120 < 2n \implies 60 < n \). Combined with the condition, this gives \( 60 < n < 100 \).
  • If \( n \ge 100 \): \( (n - 100) < (n - 20) \implies -100 < -20 \). This is true for all \( n \ge 100 \).

Therefore, the solution to the second inequality is \( n > 60 \). The integers in this range are \( 61, 62, \dots \).

Finding the Intersection

We need to find the integers \(n\) that satisfy both \( n < 80 \) and \( n > 60 \). This means \( 60 < n < 80 \). The integers in this range are 61, 62, ..., 79. The number of integers is \( 79 - 61 + 1 = 19 \).

Final Answer:

The number of integers satisfying both inequalities is 19.

Conclusion:

The correct option is (D): \( \boxed{19} \).

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