The number of elements in the set $$\{n \in \mathbb{Z}: |n^2 - 10n + 19| < 6\}$$ is _______.

To find the number of integers $$n$$ that satisfy $$|n^2 - 10n + 19| < 6$$, we break the absolute value inequality into two parts:

1. Solve the Inequality

The expression $$|f(x)| < 6$$ is equivalent to $$-6 < f(x) < 6$$.

Part A: $$n^2 - 10n + 19 < 6$$

$$n^2 - 10n + 13 < 0$$

Using the quadratic formula for $$n^2 - 10n + 13 = 0$$:

$$n = \frac{10 \pm \sqrt{100 - 52}}{2} = \frac{10 \pm \sqrt{48}}{2} = 5 \pm 2\sqrt{3}$$

Since $$2\sqrt{3} \approx 3.46$$:

• $$n \in (5 - 3.46, 5 + 3.46) \implies n \in (1.54, 8.46)$$
• Integers in this range: $$\{2, 3, 4, 5, 6, 7, 8\}$$

Part B: $$n^2 - 10n + 19 > -6$$

$$n^2 - 10n + 25 > 0$$

$$(n - 5)^2 > 0$$

A squared number is always greater than zero unless the base is zero.

• $$(n - 5)^2 = 0$$ when $$n = 5$$.
• Therefore, this condition is true for all integers except $$n = 5$$.

2. Combine the Conditions

We need integers that are in the set from Part A and satisfy Part B:

$$\{2, 3, 4, 5, 6, 7, 8\} \setminus \{5\}$$

The resulting set of integers is:

$$\{2, 3, 4, 6, 7, 8\}$$

3. Final Count

Counting the elements in the set: $$2, 3, 4, 6, 7, 8$$.

There are 6 elements.

Final Answer: 6