To solve the equation:
[tex]\[
\frac{x+2}{3x} - \frac{1}{x-2} = \frac{x-3}{3x}
\][/tex]
we start by simplifying each side of the equation.
1. Combine the terms on the left side into a single fraction:
[tex]\[
\frac{x+2}{3x} - \frac{1}{x-2}
\][/tex]
To combine these fractions, we need a common denominator. The common denominator for [tex]\( \frac{x+2}{3x} \)[/tex] and [tex]\( \frac{1}{x-2} \)[/tex] is [tex]\( 3x(x-2) \)[/tex].
- Adjust the fractions to have a common denominator:
[tex]\[
\frac{(x+2)(x-2)}{3x(x-2)} - \frac{3x}{3x(x-2)}
\][/tex]
Notice that [tex]\( \frac{x-3}{3x} \)[/tex] on the right already has the denominator [tex]\( 3x \)[/tex], we will write it out for simplification.
2. Simplify the numerators:
- For the first term [tex]\( \frac{x+2}{3x} \)[/tex]:
[tex]\[
\frac{(x+2)(x-2)}{3x(x-2)} = \frac{x^2 - 4}{3x(x-2)}
\][/tex]
- For the second term [tex]\( \frac{3}{3x} \)[/tex]:
[tex]\[
\frac{3}{x-2} = \frac{3x}{3x(x-2)}
\][/tex]
So the left-hand side rewrites as:
[tex]\[
\frac{x^2 - 4 - 3x}{3x(x-2)}
\][/tex]
3. Combining and simplifying the expressions:
Combine the terms over the common denominator:
[tex]\[
\frac{x^2 - 4 - 3x}{3x(x-2)} = \frac{x^2 - 3x - 4}{3x(x-2)}
\][/tex]
Likewise, the right-hand side is:
[tex]\[
\frac{x-3}{3x}
\][/tex]
Thus, we have:
[tex]\[
\frac{x^2 - 3x - 4}{3x(x-2)} = \frac{x-3}{3x}
\][/tex]
Multiplying both sides of the equation by [tex]\( 3x(x-2) \)[/tex] to eliminate the fractions:
[tex]\[
x^2 - 3x - 4 = (x-3)(x-2)
\][/tex]
4. Expand and simplify the equation:
Expand the right-hand side:
[tex]\[
x^2 - 3x - 4 = x^2 - 5x + 6
\][/tex]
Subtract [tex]\( x^2 \)[/tex] from both sides:
[tex]\[
-3x - 4 = -5x + 6
\][/tex]
Isolate [tex]\( x \)[/tex]:
First, add [tex]\( 5x \)[/tex] to both sides:
[tex]\[
2x - 4 = 6
\][/tex]
Then add 4 to both sides:
[tex]\[
2x = 10
\][/tex]
Finally, divide both sides by 2:
[tex]\[
x = 5
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
\boxed{5}
\][/tex]
