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

To find the number of integers that satisfy the given inequalities, let's break down the problem step by step.
The inequalities are:

1. |n - 60| < |n - 100|
2. |n - 100| < |n - 20|

Let's analyze each inequality separately:

1. |n - 60| < |n - 100|

To understand when this inequality is true, we need to consider the cases where the absolute values change. 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.

Now, let's move on to the second inequality:

1. |n - 100| < |n - 20|

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

So, the correct option is (D): 19

Answer 2

Given that,
\( 60<∣𝑛−100∣<∣𝑛−20∣ \)
Here, the discrepancy within the absolute value denotes the distance of \(n\) from \(60, 100\) and \(20\) on the number line. Consequently, when the absolute difference from a number is greater, \(n\) would be farther away from that number.
For example: the absolute difference between \(n\) and \(60\) is smaller than that of the absolute difference between \(n\) and \(20\). Consequently, \(n\) cannot be \(≤ 40\), as it would be closer to \(20\) than \(60\). Proximity on the number line suggests a lesser value of absolute difference. Hence, the stipulation \(n>40\) arises.
Similarly, the absolute difference between \(n\) and \(100\) is smaller than that of the absolute difference between \(n\) and \(20\). Thus, \(n\) cannot be \(60\), as it would be closer to \(20\) than \(100\). Therefore, the condition \(n>60\) emerges.
Moreover, the absolute difference between \(n\) and \(60\) is smaller than that of the absolute difference between \(n\) and \(100\). Hence, \(n\) cannot be \(> 80\), as it would be closer to \(100\) than \(60\). This gives rise to the condition \(n<80\).
The numbers that fulfill these conditions are \(61, 62, 63, 64......79\). Thus a total of \(19\) numbers.

So, the correct option is (D): \(19\)