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]\(\frac{1}{2} x=-\frac{1}{2}\)[/tex] results. What is the value of [tex]\(x\)[/tex]?

A. [tex]\(-1\)[/tex]
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 start by simplifying both sides and solving for [tex]\(x\)[/tex]:

1. Expand and simplify both sides:

The left side:
[tex]\[
\frac{1}{2}(x-14) + 11 = \frac{1}{2}x - 7 + 11 = \frac{1}{2}x + 4
\][/tex]

The right side:
[tex]\[
\frac{1}{2}x - (x-4) = \frac{1}{2}x - x + 4 = -\frac{1}{2}x + 4
\][/tex]

After simplifying both sides, we have:
[tex]\[
\frac{1}{2}x + 4 = -\frac{1}{2}x + 4
\][/tex]

2. Subtract 4 from both sides to isolate [tex]\(x\)[/tex] terms:

[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]

3. Combine like terms:

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

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

Which simplifies to:
[tex]\[
x = -1
\][/tex]

Therefore, the value of [tex]\(x\)[/tex] is [tex]\(-1\)[/tex].