Answer :
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], we will proceed with the following steps:
1. Distribute the constants on both sides:
On the left side:
[tex]\[ \frac{2}{3} \left(\frac{1}{2} x + 12\right) = \frac{2}{3} \cdot \frac{1}{2} x + \frac{2}{3} \cdot 12 \][/tex]
Simplify:
[tex]\[ \frac{2}{3} \cdot \frac{1}{2} x = \frac{2}{6} x = \frac{1}{3} x \][/tex]
[tex]\[ \frac{2}{3} \cdot 12 = 8 \][/tex]
So the left side becomes:
[tex]\[ \frac{1}{3} x + 8 \][/tex]
On the right side:
[tex]\[ \frac{1}{2}\left(\frac{1}{3} x + 14\right) - 3 = \frac{1}{2} \cdot \frac{1}{3} x + \frac{1}{2} \cdot 14 - 3 \][/tex]
Simplify:
[tex]\[ \frac{1}{2} \cdot \frac{1}{3} x = \frac{1}{6} x \][/tex]
[tex]\[ \frac{1}{2} \cdot 14 = 7 \][/tex]
So the right side becomes:
[tex]\[ \frac{1}{6} x + 7 - 3 \][/tex]
Simplify:
[tex]\[ \frac{1}{6} x + 4 \][/tex]
2. Set the simplified expressions equal to each other:
[tex]\[ \frac{1}{3} x + 8 = \frac{1}{6} x + 4 \][/tex]
3. Get all the x-terms on one side and the constants on the other side:
Subtract [tex]\(\frac{1}{6} x\)[/tex] from both sides:
[tex]\[ \frac{1}{3} x - \frac{1}{6} x + 8 = 4 \][/tex]
4. Combine like terms:
Note that:
[tex]\[ \frac{1}{3} x - \frac{1}{6} x = \frac{2}{6} x - \frac{1}{6} x = \frac{1}{6} x \][/tex]
So we have:
[tex]\[ \frac{1}{6} x + 8 = 4 \][/tex]
5. Isolate the [tex]\(x\)[/tex]-term:
Subtract 8 from both sides:
[tex]\[ \frac{1}{6} x = 4 - 8 \][/tex]
Simplify:
[tex]\[ \frac{1}{6} x = -4 \][/tex]
6. Solve for [tex]\(x\)[/tex]:
Multiply both sides by 6:
[tex]\[ x = -4 \times 6 \][/tex]
[tex]\[ x = -24 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies the equation is [tex]\( -24 \)[/tex]. Thus, the correct answer is:
[tex]\[ -24 \][/tex]
1. Distribute the constants on both sides:
On the left side:
[tex]\[ \frac{2}{3} \left(\frac{1}{2} x + 12\right) = \frac{2}{3} \cdot \frac{1}{2} x + \frac{2}{3} \cdot 12 \][/tex]
Simplify:
[tex]\[ \frac{2}{3} \cdot \frac{1}{2} x = \frac{2}{6} x = \frac{1}{3} x \][/tex]
[tex]\[ \frac{2}{3} \cdot 12 = 8 \][/tex]
So the left side becomes:
[tex]\[ \frac{1}{3} x + 8 \][/tex]
On the right side:
[tex]\[ \frac{1}{2}\left(\frac{1}{3} x + 14\right) - 3 = \frac{1}{2} \cdot \frac{1}{3} x + \frac{1}{2} \cdot 14 - 3 \][/tex]
Simplify:
[tex]\[ \frac{1}{2} \cdot \frac{1}{3} x = \frac{1}{6} x \][/tex]
[tex]\[ \frac{1}{2} \cdot 14 = 7 \][/tex]
So the right side becomes:
[tex]\[ \frac{1}{6} x + 7 - 3 \][/tex]
Simplify:
[tex]\[ \frac{1}{6} x + 4 \][/tex]
2. Set the simplified expressions equal to each other:
[tex]\[ \frac{1}{3} x + 8 = \frac{1}{6} x + 4 \][/tex]
3. Get all the x-terms on one side and the constants on the other side:
Subtract [tex]\(\frac{1}{6} x\)[/tex] from both sides:
[tex]\[ \frac{1}{3} x - \frac{1}{6} x + 8 = 4 \][/tex]
4. Combine like terms:
Note that:
[tex]\[ \frac{1}{3} x - \frac{1}{6} x = \frac{2}{6} x - \frac{1}{6} x = \frac{1}{6} x \][/tex]
So we have:
[tex]\[ \frac{1}{6} x + 8 = 4 \][/tex]
5. Isolate the [tex]\(x\)[/tex]-term:
Subtract 8 from both sides:
[tex]\[ \frac{1}{6} x = 4 - 8 \][/tex]
Simplify:
[tex]\[ \frac{1}{6} x = -4 \][/tex]
6. Solve for [tex]\(x\)[/tex]:
Multiply both sides by 6:
[tex]\[ x = -4 \times 6 \][/tex]
[tex]\[ x = -24 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies the equation is [tex]\( -24 \)[/tex]. Thus, the correct answer is:
[tex]\[ -24 \][/tex]
