Solveeit Logo
Login

Question

Quantitative Aptitude Question on Linear Inequalities

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

19

Explanation

Solution

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