To solve for the value of [tex]\( x \)[/tex] given the table and the mean, let's break down the solution step-by-step:
1. Identify Given Data: The table provides us with two rows of data. The first row is a series of fixed numbers, and the second row contains expressions involving [tex]\( x \)[/tex].
[tex]\[
\begin{array}{c|c|c|c|c}
x & 2 & 3 & 4 & 5 \\
\hline
x+2 & x-1/2 & x-3 & x+4 & 3x-4
\end{array}
\][/tex]
2. List All Values:
- Fixed values: [tex]\( 2, 3, 4, 5 \)[/tex]
- Expressions involving [tex]\( x \)[/tex]: [tex]\( x-1/2, x-3, x+4, 3x-4 \)[/tex]
3. Compute the Total Sum:
Adding up all elements, we get:
[tex]\[
\text{Total Sum} = 2 + 3 + 4 + 5 + \left(x - \frac{1}{2}\right) + (x - 3) + (x + 4) + (3x - 4)
\][/tex]
4. Simplify the Sum:
Combine the constants and the terms involving [tex]\( x \)[/tex]:
[tex]\[
\text{Sum of constants} = 2 + 3 + 4 + 5 - \frac{1}{2} - 3 + 4 - 4 = 10.5
\][/tex]
[tex]\[
\text{Sum of terms with } x = x + x + x + 3x = 6x
\][/tex]
Combining these, the total sum is:
[tex]\[
\text{Total Sum} = 6x + 10.5
\][/tex]
5. Number of Elements:
There are 4 fixed numbers and 4 expressions, giving us a total of 8 elements.
6. Given Mean:
It is provided that the mean is [tex]\( \frac{43}{14} \)[/tex].
7. Set Up the Mean Equation:
[tex]\[
\frac{\text{Total Sum}}{\text{Number of Elements}} = \text{Mean}
\][/tex]
Substitute the known values:
[tex]\[
\frac{6x + 10.5}{8} = \frac{43}{14}
\][/tex]
8. Solve the Equation:
Multiply both sides by 8 to clear the denominator:
[tex]\[
6x + 10.5 = 8 \times \frac{43}{14}
\][/tex]
9. Simplify the Right Side:
[tex]\[
8 \times \frac{43}{14} = \frac{8 \times 43}{14} = \frac{344}{14} = 24.571428571428573
\][/tex]
So,
[tex]\[
6x + 10.5 = 24.571428571428573
\][/tex]
10. Isolate [tex]\( x \)[/tex]:
Subtract 10.5 from both sides:
[tex]\[
6x = 24.571428571428573 - 10.5
\][/tex]
[tex]\[
6x = 14.071428571428573
\][/tex]
Divide both sides by 6:
[tex]\[
x = \frac{14.071428571428573}{6} = 2.34523809523810
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is approximately [tex]\( 2.34523809523810 \)[/tex].
