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 carefully follow the steps taken by Karissa:

1. Expand both sides of the equation:

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

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

2. Simplify both sides:

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

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

3. Equating the simplified expressions:

- Now, the equation [tex]\(\frac{1}{2}x + 4 = -\frac{1}{2}x + 4\)[/tex] results.

4. Subtract 4 from both sides:

- Doing this, we get: [tex]\(\frac{1}{2}x = -\frac{1}{2}x\)[/tex].

5. Solve for [tex]\(x\)[/tex]:

- Add [tex]\(\frac{1}{2}x\)[/tex] to both sides to combine the [tex]\(x\)[/tex] terms: [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 0.