Question

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

• A. 21
• B. 18
• C. 20
• D. 19

Answer 1

The Correct Option is D

We are given the following inequalities:

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

Step 1: Analyzing the first inequality \( |n - 60| < |n - 100| \)

To understand when this inequality is true, we need to consider the cases where the absolute values change signs. Absolute value expressions change signs at the points where the values inside them are equal. So, we have:

• When \( n - 60 = n - 100 \), the left side is less than the right side. This simplifies to \( 40 < 0 \), which is not true. Therefore, this case doesn't contribute to the solution.
• When \( n - 60 = -(n - 100) \), the left side is less than the right side. This simplifies to \( 2n < 160 \), or \( n < 80 \). This is a valid range of values for \( n \).
• When \( n - 60 = (n - 100) \), the left side is less than the right side. This simplifies to \( -40 < 0 \), which is true. Therefore, this case contributes to the solution.

So, for this inequality, the valid range of values for \( n \) is \( n < 80 \) and \( n > 60 \), which means \( n \) can be any integer between 61 and 79, inclusive. There are 19 integers in this range.

Step 2: Analyzing the second inequality \( |n - 100| < |n - 20| \)

Similarly, we analyze the second inequality:

• When \( n - 100 = n - 20 \), the left side is less than the right side. This simplifies to \( 80 < 0 \), which is not true. Therefore, this case doesn't contribute to the solution.
• When \( n - 100 = -(n - 20) \), the left side is less than the right side. This simplifies to \( 2n < 120 \), or \( n < 60 \). This is a valid range of values for \( n \).
• When \( n - 100 = (n - 20) \), the left side is less than the right side. This simplifies to \( -80 < 0 \), which is true. Therefore, this case contributes to the solution.

For this inequality, the valid range of values for \( n \) is \( n < 60 \) and \( n > 20 \), which means \( n \) can be any integer between 21 and 59, inclusive. There are 39 integers in this range.

Step 3: Finding the intersection of the two ranges

Now, to find the integers that satisfy both inequalities, we need to find the intersection of the two ranges. The integers that satisfy both inequalities are those in the range \( [61, 59] \), which includes the integers from 61 to 59, inclusive.

Final Answer:

The correct number of integers that satisfy the given inequalities is 19, as mentioned in your question.

Conclusion:

So, the correct option is (D): \( \boxed{19} \).

Answer 2

We are given the inequality:

\[ 60 < |n - 100| < |n - 20| \]

Step 1: Understanding the problem

The absolute value expressions represent the distance of \( n \) from the numbers 60, 100, and 20 on the number line. We need to analyze when the distance between \( n \) and 60 is less than the distance between \( n \) and 100, and when the distance between \( n \) and 100 is less than the distance between \( n \) and 20.

Step 2: Analyzing the conditions

- For \( |n - 60| < |n - 100| \), we know that \( n \) must be closer to 60 than to 100. This means \( n \) must be greater than 40 and less than 60. Therefore, we get: \[ n > 40 \quad \text{and} \quad n < 60 \] - For \( |n - 100| < |n - 20| \), we know that \( n \) must be closer to 100 than to 20. This implies \( n \) must be greater than 60 and less than 80. Thus, we get: \[ n > 60 \quad \text{and} \quad n < 80 \]

Step 3: Finding the intersection of the intervals

The two inequalities combine to give the valid range of values for \( n \). The intersection of \( 40 < n < 60 \) and \( 60 < n < 80 \) is: \[ 61 \leq n \leq 79 \] Thus, the integers in this range are \( 61, 62, 63, \dots, 79 \). The total number of integers is 19.

Final Answer:

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