Certainly! Let's solve the equation step-by-step:
Given equation:
[tex]\[
\frac{1}{3} x - 8 = \frac{1}{2}(x - 12) + 4
\][/tex]
### Step 1: Distribute on the right side
First, let's distribute the [tex]\(\frac{1}{2}\)[/tex] on the right side of the equation.
[tex]\[
\frac{1}{3} x - 8 = \frac{1}{2} x - \frac{1}{2} \cdot 12 + 4
\][/tex]
Simplifying the right side:
[tex]\[
\frac{1}{3} x - 8 = \frac{1}{2} x - 6 + 4
\][/tex]
[tex]\[
\frac{1}{3} x - 8 = \frac{1}{2} x - 2
\][/tex]
### Step 2: Move all [tex]\(x\)[/tex] terms to one side of the equation
To do this, we'll subtract [tex]\(\frac{1}{2} x\)[/tex] from both sides of the equation:
[tex]\[
\frac{1}{3} x - \frac{1}{2} x - 8 = -2
\][/tex]
### Step 3: Combine the [tex]\(x\)[/tex] terms
We need a common denominator to combine [tex]\(\frac{1}{3} x\)[/tex] and [tex]\(\frac{1}{2} x\)[/tex]. The common denominator of 3 and 2 is 6, so we rewrite the terms:
[tex]\[
\frac{2}{6} x - \frac{3}{6} x - 8 = -2
\][/tex]
Combining the [tex]\(x\)[/tex] terms:
[tex]\[
-\frac{1}{6} x - 8 = -2
\][/tex]
### Step 4: Isolate the [tex]\(x\)[/tex] term
Add 8 to both sides to move the constant term to the right side:
[tex]\[
-\frac{1}{6} x - 8 + 8 = -2 + 8
\][/tex]
[tex]\[
-\frac{1}{6} x = 6
\][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], multiply both sides by [tex]\(-6\)[/tex]:
[tex]\[
x = 6 \times -6
\][/tex]
[tex]\[
x = -36
\][/tex]
### Solution
So, the solution to the equation is:
[tex]\[
x = -36
\][/tex]
