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]

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].

The original equation given is:

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

First, let's simplify both sides:

Left Side:

Distribute [tex]\(\frac{1}{2}\)[/tex] in [tex]\(\frac{1}{2}(x-14)\)[/tex]:

[tex]\[
\frac{1}{2}x - \frac{1}{2} \times 14 = \frac{1}{2}x - 7
\][/tex]

Now, adding 11:

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

Right Side:

Distribute the negative sign in [tex]\(-(x-4)\)[/tex]:

[tex]\[
-x + 4
\][/tex]

The equation becomes:

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

Simplify the right side:

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

So the equation is now:

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

Next, subtract 4 from both sides:

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

Add [tex]\(\frac{1}{2}x\)[/tex] to both sides:

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

This simplifies to:

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

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