College

Karissa begins to solve the equation [tex]\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)[/tex]. Her work is correct and is shown below.

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

When she subtracts 4 from both sides, [tex]\frac{1}{2} x=-\frac{1}{2} x[/tex] results. What is the value of [tex]x[/tex]?

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

Answer :

We start with the equation

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

1. Distribute and simplify each side:
On the left, distribute the [tex]$\frac{1}{2}$[/tex] over [tex]$(x-14)$[/tex]:

[tex]$$
\frac{1}{2}(x-14)=\frac{1}{2}x-\frac{1}{2}\cdot14=\frac{1}{2}x-7.
$$[/tex]

Then, add [tex]$11$[/tex] to obtain:

[tex]$$
\frac{1}{2}x-7+11 = \frac{1}{2}x+4.
$$[/tex]

On the right, distribute the negative sign:

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

Simplify the [tex]$\frac{1}{2}x - x$[/tex] by writing [tex]$x$[/tex] as [tex]$\frac{2}{2}x$[/tex]:

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

so the right side becomes:

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

2. Write the simpler equation:
The equation now is:

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

3. Subtract [tex]$4$[/tex] from both sides:
Eliminating the constant term, we subtract [tex]$4$[/tex]:

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

which simplifies to:

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

4. Combine like terms:
Add [tex]$\frac{1}{2}x$[/tex] to both sides to bring like terms together:

[tex]$$
\frac{1}{2}x+\frac{1}{2}x = 0.
$$[/tex]

The left side becomes:

[tex]$$
x = 0.
$$[/tex]

Thus, the value of [tex]$x$[/tex] is

[tex]$$
\boxed{0}.
$$[/tex]