College

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

Answer :

We start with the equation

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

Step 1. Expand both sides:

On the left side:

[tex]$$
\frac{1}{2}(x-14) + 11 = \frac{1}{2}x - 7 + 11 = \frac{1}{2}x + 4.
$$[/tex]

On the right side:

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

Now the equation becomes

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

Step 2. Subtract 4 from both sides to isolate the terms with [tex]$x$[/tex]:

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

which simplifies to

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

Step 3. Add [tex]$\frac{1}{2} x$[/tex] to both sides to combine like terms:

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

which simplifies to

[tex]$$
x = 0.
$$[/tex]

Thus, the value of [tex]$x$[/tex] is [tex]$0$[/tex].

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