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 :

Sure, let's work through the equation step-by-step to find the value of [tex]\( x \)[/tex].

We start with the equation Karissa was solving:

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

Step 1: Simplify both sides.

- Distribute on the left side:
[tex]\[
\frac{1}{2}(x-14) = \frac{1}{2}x - 7
\][/tex]
So the left side becomes:
[tex]\[
\frac{1}{2}x - 7 + 11
\][/tex]
Simplify the constants:
[tex]\[
\frac{1}{2}x + 4
\][/tex]

- On the right side, distribute the negative sign:
[tex]\[
-(x-4) = -x + 4
\][/tex]
Substituting it back, the right side becomes:
[tex]\[
\frac{1}{2}x - x + 4
\][/tex]

Step 2: Set them equal (as shown after simplification in the solution):
[tex]\[
\frac{1}{2}x + 4 = -\frac{1}{2}x + 4
\][/tex]

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

Step 4: Combine like terms by adding [tex]\(\frac{1}{2}x\)[/tex] to both sides to eliminate the negative term:
[tex]\[
\frac{1}{2}x + \frac{1}{2}x = 0
\][/tex]

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

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