The number of elements in the set {n in mathb | vedclass

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(D) The given inequality is $|n^2 - 10n + 19| < 6$.This is equivalent to $-6 < n^2 - 10n + 19 < 6$.Case $1$: $n^2 - 10n + 19 < 6 \Rightarrow n^2 - 10n

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$6$

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(D) The given inequality is $|n^2 - 10n + 19| This is equivalent to $-6 Case $1$: $n^2 - 10n + 19 The roots of $n^2 - 10n + 13 = 0$ are $n = \frac{10 \pm \sqrt{100 - 52}}{2} = 5 \pm \sqrt{12} = 5 \pm 2\sqrt{3}$.Since $2\sqrt{3} \approx 3.46$,the range is $n \in (5 - 3.46, 5 + 3.46) = (1.54, 8.46)$.Case $2$: $n^2 - 10n + 19 > -6$ $\Rightarrow n^2 - 10n + 25 > 0$ $\Rightarrow (n - 5)^2 > 0$.This is true for all $n \in \mathbb{Z}$ except $n = 5$.Combining these,$n \in \{2, 3, 4, 6, 7, 8\}$.The number of such elements is $6$.

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

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