To solve the equation [tex]\(\frac{2}{3}\left(\frac{1}{2} x+12\right) = \frac{1}{2}\left(\frac{1}{3} x+14\right) - 3\)[/tex], let's go through it step-by-step.
1. Expand both sides:
For the left side:
[tex]\[
\frac{2}{3} \left(\frac{1}{2} x + 12\right) = \frac{2}{3} \cdot \frac{1}{2} x + \frac{2}{3} \cdot 12 = \frac{1}{3} x + 8
\][/tex]
For the right side:
[tex]\[
\frac{1}{2} \left(\frac{1}{3} x + 14\right) - 3 = \frac{1}{2} \cdot \frac{1}{3} x + \frac{1}{2} \cdot 14 - 3 = \frac{1}{6} x + 7 - 3 = \frac{1}{6} x + 4
\][/tex]
2. Set up the equation with the expanded expressions:
[tex]\[
\frac{1}{3} x + 8 = \frac{1}{6} x + 4
\][/tex]
3. Eliminate the fractions by finding a common multiple:
The least common multiple of 3 and 6 is 6. Multiply both sides of the equation by 6 to clear the fractions:
[tex]\[
6 \left(\frac{1}{3} x + 8\right) = 6 \left(\frac{1}{6} x + 4\right)
\][/tex]
Simplifying this:
[tex]\[
2x + 48 = x + 24
\][/tex]
4. Isolate [tex]\(x\)[/tex] on one side of the equation:
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
2x - x + 48 = 24
\][/tex]
Simplifying:
[tex]\[
x + 48 = 24
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Subtract 48 from both sides:
[tex]\[
x = 24 - 48
\][/tex]
Simplifying:
[tex]\[
x = -24
\][/tex]
Thus, the value of [tex]\(x\)[/tex] is [tex]\(-24\)[/tex].
The correct answer is:
[tex]\[
-24
\][/tex]
