To solve the equation [tex]\(\frac{1}{3}(x-2) = \frac{1}{5}(x+4) + 2\)[/tex], we will follow these steps:
1. Remove the fractions by finding a common denominator.
First, multiply both sides of the equation by the least common multiple (LCM) of the denominators, which are 3 and 5. The LCM of 3 and 5 is 15.
[tex]\[
15 \cdot \frac{1}{3}(x-2) = 15 \cdot \left( \frac{1}{5}(x+4) + 2 \right)
\][/tex]
2. Distribute the 15 into each term.
[tex]\[
15 \cdot \frac{1}{3}(x-2) = 15 \cdot \frac{1}{3} \cdot (x-2) = 5(x - 2)
\][/tex]
[tex]\[
15 \cdot \left( \frac{1}{5}(x+4) + 2 \right) = 15 \cdot \frac{1}{5}(x+4) + 15 \cdot 2 = 3(x + 4) + 30
\][/tex]
So, substituting back we get:
[tex]\[
5(x - 2) = 3(x + 4) + 30
\][/tex]
3. Simplify the equation.
First, distribute the constants on both sides:
[tex]\[
5x - 10 = 3x + 12 + 30
\][/tex]
Combine like terms on the right side:
[tex]\[
5x - 10 = 3x + 42
\][/tex]
4. Isolate the variable [tex]\(x\)[/tex].
Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
5x - 3x - 10 = 3x - 3x + 42
\][/tex]
[tex]\[
2x - 10 = 42
\][/tex]
Add 10 to both sides:
[tex]\[
2x - 10 + 10 = 42 + 10
\][/tex]
[tex]\[
2x = 52
\][/tex]
5. Solve for [tex]\(x\)[/tex].
Divide both sides by 2:
[tex]\[
x = \frac{52}{2}
\][/tex]
[tex]\[
x = 26
\][/tex]
So, the solution to the equation [tex]\(\frac{1}{3}(x-2) = \frac{1}{5}(x+4) + 2\)[/tex] is [tex]\( \boxed{26} \)[/tex].
