High School

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

Answer :

Let's solve the equation step by step to find the value of [tex]\( x \)[/tex].

We're starting with the equation:
[tex]\[
\frac{1}{2}(x-14) + 11 = \frac{1}{2}x - (x-4)
\][/tex]

1. Distribute and simplify:

Distribute [tex]\(\frac{1}{2}\)[/tex] on the left side:
[tex]\[
\frac{1}{2}x - 7 + 11
\][/tex]

Simplify it:
[tex]\[
\frac{1}{2}x + 4
\][/tex]

Distribute the minus sign on the right side:
[tex]\[
\frac{1}{2}x - x + 4
\][/tex]

2. Set the simplified forms equal:

Now we have:
[tex]\[
\frac{1}{2}x + 4 = -\frac{1}{2}x + 4
\][/tex]

3. 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]

4. Eliminate [tex]\(-\frac{1}{2}x\)[/tex] from the right side by adding it to both sides:

Adding [tex]\(\frac{1}{2}x\)[/tex] to both sides:
[tex]\[
\frac{1}{2}x + \frac{1}{2}x = 0
\][/tex]

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

So, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{0}\)[/tex].