## Answer :

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].