Certainly! Let's walk through the solution step by step.
We are given five data values:
[tex]\[ x + 4, \quad 2x + 2, \quad 5x, \quad 4x + 1, \quad 6x + 2 \][/tex]
We know that the mean of these data values is 12, and we need to find the value of [tex]\( x \)[/tex] that satisfies this condition.
### Step 1: Calculate the Mean of the Data Values
The mean (average) of a set of values is calculated by summing all the values and then dividing by the number of values.
Let's denote our five data values as [tex]\( a_1, a_2, a_3, a_4, a_5 \)[/tex]. So:
[tex]\[
a_1 = x + 4, \quad a_2 = 2x + 2, \quad a_3 = 5x, \quad a_4 = 4x + 1, \quad a_5 = 6x + 2
\][/tex]
Sum of these data values:
[tex]\[
(x + 4) + (2x + 2) + (5x) + (4x + 1) + (6x + 2)
\][/tex]
Simplifying the summation:
[tex]\[
x + 4 + 2x + 2 + 5x + 4x + 1 + 6x + 2 = 18x + 9
\][/tex]
There are 5 values, so the mean is:
[tex]\[
\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}} = \frac{18x + 9}{5}
\][/tex]
### Step 2: Set Up the Equation
We are given that the mean is 12:
[tex]\[
\frac{18x + 9}{5} = 12
\][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
We need to solve the following equation:
[tex]\[
\frac{18x + 9}{5} = 12
\][/tex]
To isolate [tex]\( 18x + 9 \)[/tex], multiply both sides by 5:
[tex]\[
18x + 9 = 60
\][/tex]
Next, isolate the term with [tex]\( x \)[/tex] by subtracting 9 from both sides:
[tex]\[
18x = 51
\][/tex]
Finally, solve for [tex]\( x \)[/tex] by dividing both sides by 18:
[tex]\[
x = \frac{51}{18} = \frac{17}{6}
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that makes the mean of the given data values equal to 12 is:
[tex]\[
x = \frac{17}{6}
\][/tex]
