Question

The number of elements in the set \(\{n ∈ Z : n^2-10n+19| < 6\}\) is __.

Answer 1

Correct Answer: 6

 To determine the number of elements in the set \( \{ n \in \mathbb{Z} : |n^2 - 10n + 19| < 6 \} \), we need to solve the inequality \( |n^2 - 10n + 19| < 6 \). This can be broken down into two separate inequalities:
1. \( n^2 - 10n + 19 < 6 \)
2. \( n^2 - 10n + 19 > -6 \)
Let's solve each inequality:
1. Simplifying: 
\( n^2 - 10n + 19 < 6 \) simplifies to \( n^2 - 10n + 13 < 0 \).
Factor the quadratic: \( n^2 - 10n + 13 = (n-5)^2 - 12 \).
We need \( (n-5)^2 < 12 \).
This means \( -\sqrt{12} < n-5 < \sqrt{12} \). Approximating \( \sqrt{12} \approx 3.46 \), so:
\(-3.46 < n-5 < 3.46 \) simplifies to \( 1.54 < n < 8.46 \).
Thus, \( n \) is an integer in the range \( [2, 8] \).
2. Simplifying: 
\( n^2 - 10n + 19 > -6 \) simplifies to \( n^2 - 10n + 25 > 0 \).
This is equivalent to \( (n-5)^2 > 0 \).
\( (n-5)^2 > 0 \) holds true for all integers \( n \neq 5 \).
Combining both conditions:
The solution to both inequalities is integers such that \( 2 \leq n \leq 8 \) except \( n \neq 5 \).
The integers are \( n = 2, 3, 4, 6, 7, 8 \).
Thus, there are 6 elements in the set.
Verification:
The computed number of elements (6) matches the expected range \([6,6]\). Therefore, the set indeed contains 6 elements.