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.

[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, [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]

Step 1. Expand the expressions.

For the left side, distribute [tex]$\frac{1}{2}$[/tex]:
[tex]$$
\frac{1}{2}(x-14)=\frac{1}{2}x-\frac{1}{2}\times14=\frac{1}{2}x-7.
$$[/tex]
So, the left side becomes:
[tex]$$
\frac{1}{2}x-7+11=\frac{1}{2}x+4.
$$[/tex]

For the right side, distribute the negative sign:
[tex]$$
\frac{1}{2}x-(x-4)=\frac{1}{2}x-x+4=-\frac{1}{2}x+4.
$$[/tex]

The equation now is:
[tex]$$
\frac{1}{2}x+4=-\frac{1}{2}x+4.
$$[/tex]

Step 2. Subtract 4 from both sides.

Subtract [tex]$4$[/tex] from each side to eliminate the constant:
[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].

To isolate [tex]$x$[/tex], add [tex]$\frac{1}{2}x$[/tex] to both sides:
[tex]$$
\frac{1}{2}x + \frac{1}{2}x= -\frac{1}{2}x+\frac{1}{2}x.
$$[/tex]
This gives:
[tex]$$
x=0.
$$[/tex]

Thus, the value of [tex]$x$[/tex] is [tex]$\boxed{0}$[/tex].