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 the steps in detail:
1. Distribute the constants inside the parentheses:
[tex]\[
\frac{2}{3} \left( \frac{1}{2} x + 12 \right) = \frac{1}{2} \left( \frac{1}{3} x + 14 \right) - 3
\][/tex]
[tex]\[
\frac{2}{3} \cdot \frac{1}{2} x + \frac{2}{3} \cdot 12 = \frac{1}{2} \cdot \frac{1}{3} x + \frac{1}{2} \cdot 14 - 3
\][/tex]
2. Simplify the multiplications:
[tex]\[
\frac{1}{3} x + 8 = \frac{1}{6} x + 7 - 3
\][/tex]
[tex]\[
\frac{1}{3} x + 8 = \frac{1}{6} x + 4
\][/tex]
3. Eliminate the fractions by finding a common denominator. The common denominator for 3 and 6 is 6. Multiply every term by 6:
[tex]\[
6 \left( \frac{1}{3} x \right) + 6 \cdot 8 = 6 \left( \frac{1}{6} x \right) + 6 \cdot 4
\][/tex]
[tex]\[
2x + 48 = x + 24
\][/tex]
4. Isolate the variable [tex]\( x \)[/tex]:
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
2x - x + 48 = 24
\][/tex]
[tex]\[
x + 48 = 24
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Subtract 48 from both sides:
[tex]\[
x = 24 - 48
\][/tex]
[tex]\[
x = -24
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\(-24\)[/tex]. Therefore, the correct answer is [tex]\(-24\)[/tex].
