The number of elements in the set \(\{n ∈ Z : n^2-10n+19| < 6\}\) is __.
Correct Answer: 6
Solution and Explanation
Given the inequality: \[ -6 < n^2 - 10n + 19 < 6 \]
Step 1: Split the Inequality
Split the inequality into two parts:
- \( n^2 - 10n + 19 - 6 > 0 \) and
- \( n^2 - 10n + 19 + 6 < 0 \)
This simplifies to:
- \( n^2 - 10n + 25 > 0 \)
- \( n^2 - 10n + 13 < 0 \)
Step 2: Solve \( n^2 - 10n + 25 > 0 \)
Factorize:
\[ (n - 5)^2 > 0 \]
The solution is \( n \in \mathbb{Z} \setminus \{5\} \) (all integers except 5). Call this result (i).
Step 3: Solve \( n^2 - 10n + 13 < 0 \)
Find the roots of the equation:
\[ n^2 - 10n + 13 = 0 \]
The roots are:
\[ n = 5 \pm \sqrt{3} \]
Approximating the roots:
\[ 5 - \sqrt{3} \approx 2 \quad \text{and} \quad 5 + \sqrt{3} \approx 8 \]
Thus, \( 2 < n < 8 \). For integer values, \( n \in \{2, 3, 4, 5, 6, 7, 8\} \). Call this result (ii).
Step 4: Combine Results
From (i) and (ii):
\( n \in \{2, 3, 4, 5, 6, 8\} \).
Final Answer:
The number of values of \( n \) is:
\[ \boxed{6} \]
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