To find the unknown value [tex]\( x \)[/tex] in the dataset [tex]\(\{3, 5, 8, x, 6, 4, 7, 9\}\)[/tex] given that the arithmetic mean of the data is 7, follow the steps below:
1. Sum up the known values in the dataset:
Add the numbers [tex]\(3, 5, 8, 6, 4, 7,\)[/tex] and [tex]\(9\)[/tex]:
[tex]\[
3 + 5 + 8 + 6 + 4 + 7 + 9 = 42
\][/tex]
2. Determine the total number of data points in the dataset:
The dataset includes [tex]\(8\)[/tex] numbers: [tex]\(\{3, 5, 8, x, 6, 4, 7, 9\}\)[/tex].
3. Use the formula for the arithmetic mean:
The arithmetic mean is given by:
[tex]\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Total number of data points}}
\][/tex]
Here, the mean is [tex]\(7\)[/tex] and the total number of data points is [tex]\(8\)[/tex]. Therefore:
[tex]\[
\text{Sum of all data points} = 7 \times 8 = 56
\][/tex]
4. Set up the equation to find the unknown value [tex]\( x \)[/tex]:
Let [tex]\( S \)[/tex] represent the sum of all the given data points, including [tex]\( x \)[/tex]. We know:
[tex]\[
S = 42 + x \quad \text{and} \quad S = 56
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Set the equation:
[tex]\[
42 + x = 56
\][/tex]
Subtract [tex]\(42\)[/tex] from both sides of the equation to isolate [tex]\( x \)[/tex]:
[tex]\[
x = 56 - 42
\][/tex]
[tex]\[
x = 14
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{14} \)[/tex].
So, the correct answer is [tex]\( c. 14 \)[/tex].
