Question

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

Answer 1

Correct Answer: 6

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:

\[ \boxed{6} \]