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

Answer :

To solve the equation [tex]\(\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)\)[/tex], let's go through the steps one by one:

1. Distribute and simplify both sides:
- On the left side, distribute [tex]\(\frac{1}{2}\)[/tex] into [tex]\((x-14)\)[/tex]:
[tex]\[
\frac{1}{2} x - 7 + 11
\][/tex]
This simplifies to:
[tex]\[
\frac{1}{2} x + 4
\][/tex]

- On the right side, distribute the negative sign:
[tex]\[
\frac{1}{2} x - x + 4
\][/tex]
This simplifies to:
[tex]\[
-\frac{1}{2} x + 4
\][/tex]

2. Set the simplified expressions equal to each other:
[tex]\[
\frac{1}{2} x + 4 = -\frac{1}{2} x + 4
\][/tex]

3. Subtract 4 from both sides:
[tex]\[
\frac{1}{2} x = -\frac{1}{2} x
\][/tex]

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

5. Combine like terms on the left side:
[tex]\[
x = 0
\][/tex]

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