Thanks to pointed out. Many thanks.

Little plot twist: if the inequality were \(|2n-7|\geq 10\), the solution would have been \(n \geq \frac{3}{2}\) and \(n \leq -\frac{17}{2}\).
In this case the difference between the highest and smallest values of n would have been 2 - (-9) = 11 so that A would have been greater!

Why quantity A is 1-(-8).. plz explain me this step.
thanks

The smallest integer value that is possible is \(-8\) and the largest integer value is 1.

So Largest -smallest=\(1- (-8)=9\)

Why not?

This cannot be correct. If n > or equal to 3/2 (1.5), then that value is endless. The highest value is infinity.
Same with the lower end value, if it's lesser than -8.5, that is -9.5 and so on... hence, least value is infinity to the left of the spectrum.

So, if we are gonna include the numbers from -8 to 1: -8 -7 -6 -5 -4 -3 -2 -1 0 1. This is 10 numbers right, not 9 numbers. When we find the number of numbers through subtraction it tends to exclude a number from the range. So, technically arent we supposed to add another number to 1-(-8) +1 [for inclusive]

So, if we are gonna include the numbers from -8 to 1: -8 -7 -6 -5 -4 -3 -2 -1 0 1. This is 10 numbers right, not 9 numbers. When we find the number of numbers through subtraction it tends to exclude a number from the range. So, technically arent we supposed to add another number to 1-(-8) +1 [for inclusive]

Hey,

The question simply asks the difference between the greatest and the least(the range) and it is \(9\) here.

When solving inequalities involving ABSOLUTE VALUE, there are 2 things you need to know:
Rule #1: If |something| < k, then –k < something < k
Rule #2: If |something| > k, then EITHER something > k OR something < -k
Note: these rules assume that k is positive

Given: |2n + 7| ≤ 10
In this case, we'll apply rule #1 to get: -10 ≤ 2n + 7 ≤ 10
Subtract 7 from all three parts: -17 ≤ 2n ≤ 3
Divide all three parts by 2 to get: -8.5 ≤ n ≤ 1.5
Since n is an INTEGER, the possible values of n are: -8, -7, -6, -5,....0, 1

The question:
Quantity A: The difference between the greatest and least possible values of n
Quantity B: 10

We get:
Quantity A: 1 - (-8)
Quantity B: 10

Evaluate:
Quantity A: 9
Quantity B: 10

Answer: B

Sir, even though I understood the problem, can you please explain how the rules I &II work with some specific examples.

Rule #1: If |something| < k, then –k < something < k
Example: If |2x + 1| < 3, then it must be true that -3 < 2x + 1 < 3
Example: If |4n| < 11.3, then it must be true that -11.3 < 4n < 11.3

Rule #2: If |something| > k, then EITHER something > k OR something < -k
Example: If |3y - 1| > 5, then it must be true that either 3y - 1 > 5 or 3y - 1 < -5
Example: If |7w| > 6.1, then it must be true that either 7w > 6.1 or 7w < -6.1

I found this easiest to simply think through the positive and negative integers that fulfill this inequality, and then to subtract the greatest and least of these.

The list I made was -8 -7 -6 -5 -4 -3 -2 -1 0 1.

The least of these is -8 and the greatest is 1. The difference between these two values is 9. Hence, B=10 is greater.

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