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Answer :
We begin with the equation
[tex]$$
\frac{1}{2}(x-14)+11=\frac{1}{2}x-(x-4).
$$[/tex]
Step 1. Expand both sides.
On the left side, distribute [tex]$\frac{1}{2}$[/tex] over [tex]$(x-14)$[/tex]:
[tex]$$
\frac{1}{2}(x-14) + 11 = \frac{1}{2}x - 7 + 11.
$$[/tex]
Combine the constant terms:
[tex]$$
\frac{1}{2}x - 7 + 11 = \frac{1}{2}x + 4.
$$[/tex]
On the right side, distribute the negative sign:
[tex]$$
\frac{1}{2}x - (x-4) = \frac{1}{2}x - x + 4.
$$[/tex]
Combine like terms for [tex]$x$[/tex]:
[tex]$$
\frac{1}{2}x - x = -\frac{1}{2}x,
$$[/tex]
so the right side becomes:
[tex]$$
-\frac{1}{2}x + 4.
$$[/tex]
Step 2. Set the simplified expressions equal:
[tex]$$
\frac{1}{2}x + 4 = -\frac{1}{2}x + 4.
$$[/tex]
Step 3. Eliminate the constant by subtracting 4 from both sides:
[tex]$$
\frac{1}{2}x = -\frac{1}{2}x.
$$[/tex]
Step 4. Solve for [tex]$x$[/tex]. Add [tex]$\frac{1}{2}x$[/tex] to both sides:
[tex]$$
\frac{1}{2}x + \frac{1}{2}x = 0 \quad \Longrightarrow \quad x = 0.
$$[/tex]
Thus, the value of [tex]$x$[/tex] is
[tex]$$
\boxed{0}.
$$[/tex]
[tex]$$
\frac{1}{2}(x-14)+11=\frac{1}{2}x-(x-4).
$$[/tex]
Step 1. Expand both sides.
On the left side, distribute [tex]$\frac{1}{2}$[/tex] over [tex]$(x-14)$[/tex]:
[tex]$$
\frac{1}{2}(x-14) + 11 = \frac{1}{2}x - 7 + 11.
$$[/tex]
Combine the constant terms:
[tex]$$
\frac{1}{2}x - 7 + 11 = \frac{1}{2}x + 4.
$$[/tex]
On the right side, distribute the negative sign:
[tex]$$
\frac{1}{2}x - (x-4) = \frac{1}{2}x - x + 4.
$$[/tex]
Combine like terms for [tex]$x$[/tex]:
[tex]$$
\frac{1}{2}x - x = -\frac{1}{2}x,
$$[/tex]
so the right side becomes:
[tex]$$
-\frac{1}{2}x + 4.
$$[/tex]
Step 2. Set the simplified expressions equal:
[tex]$$
\frac{1}{2}x + 4 = -\frac{1}{2}x + 4.
$$[/tex]
Step 3. Eliminate the constant by subtracting 4 from both sides:
[tex]$$
\frac{1}{2}x = -\frac{1}{2}x.
$$[/tex]
Step 4. Solve for [tex]$x$[/tex]. Add [tex]$\frac{1}{2}x$[/tex] to both sides:
[tex]$$
\frac{1}{2}x + \frac{1}{2}x = 0 \quad \Longrightarrow \quad x = 0.
$$[/tex]
Thus, the value of [tex]$x$[/tex] is
[tex]$$
\boxed{0}.
$$[/tex]
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