Certainly! Let's solve this step-by-step.
We have five algebraic expressions representing the data values:
1. [tex]\( x + 4 \)[/tex]
2. [tex]\( 2x + 2 \)[/tex]
3. [tex]\( 5x \)[/tex]
4. [tex]\( 4x + 1 \)[/tex]
5. [tex]\( 6x + 2 \)[/tex]
We know that the mean of these values is 12. To find the value of [tex]\( x \)[/tex], we can follow these steps:
### Step 1: Calculate the total sum of the values
First, let's sum up all the values:
[tex]\[
(x + 4) + (2x + 2) + (5x) + (4x + 1) + (6x + 2)
\][/tex]
Combining like terms:
[tex]\[
x + 2x + 5x + 4x + 6x + 4 + 2 + 1 + 2 = 18x + 9
\][/tex]
### Step 2: Use the mean to set up an equation
The mean is the total sum of the values divided by the number of values. We are given that the mean is 12. Therefore,
[tex]\[
\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Values}} = 12
\][/tex]
[tex]\[
\frac{18x + 9}{5} = 12
\][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
We need to solve this equation:
[tex]\[
\frac{18x + 9}{5} = 12
\][/tex]
Multiply both sides by 5 to clear the fraction:
[tex]\[
18x + 9 = 60
\][/tex]
Subtract 9 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
18x = 51
\][/tex]
Divide both sides by 18 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{51}{18}
\][/tex]
Simplify the fraction:
[tex]\[
x = \frac{51 \div 3}{18 \div 3} = \frac{17}{6}
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( \frac{17}{6} \)[/tex].
