Given the equation:

[tex]
\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)
[/tex]

Her work is shown below:

[tex]
\begin{array}{c}
\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4) \\
\frac{1}{2} x-7+11=\frac{1}{2} x-x+4 \\
\frac{1}{2} x+4=-\frac{1}{2} x+4
\end{array}
[/tex]

When she subtracts 4 from both sides, the equation becomes:

[tex]
\frac{1}{2} x=-\frac{1}{2} x
[/tex]

What is the value of [tex]x[/tex]?

A. [tex]-1[/tex]
B. [tex]-\frac{1}{2}[/tex]
C. [tex]0[/tex]
D. [tex]\frac{1}{2}[/tex]

Let's solve the equation step by step.

We start with the given equation:

[tex]\[
\frac{1}{2}(x - 14) + 11 = \frac{1}{2}x - (x - 4)
\][/tex]

Step 1: Simplify both sides

First, we'll expand each side.

- On the left side:
[tex]$\frac{1}{2}(x - 14) + 11$[/tex] simplifies to [tex]$\frac{1}{2}x - 7 + 11$[/tex], which further simplifies to [tex]$\frac{1}{2}x + 4$[/tex].

- On the right side:
[tex]$\frac{1}{2}x - (x - 4)$[/tex] becomes [tex]$\frac{1}{2}x - x + 4$[/tex], which simplifies to [tex]$-\frac{1}{2}x + 4$[/tex].

So now the equation looks like:

[tex]\[
\frac{1}{2}x + 4 = -\frac{1}{2}x + 4
\][/tex]

Step 2: Eliminate constant terms

To remove the constant terms, we subtract 4 from both sides:

[tex]\[
\frac{1}{2}x + 4 - 4 = -\frac{1}{2}x + 4 - 4
\][/tex]

This simplifies to:

[tex]\[
\frac{1}{2}x = -\frac{1}{2}x
\][/tex]

Step 3: Solve for [tex]$x$[/tex]

Now, we add [tex]$\frac{1}{2}x$[/tex] to both sides to isolate the variable:

[tex]\[
\frac{1}{2}x + \frac{1}{2}x = -\frac{1}{2}x + \frac{1}{2}x
\][/tex]

This becomes:

[tex]\[
x = 0
\][/tex]

Therefore, the value of [tex]$x$[/tex] is [tex]$0$[/tex].